Sufficient and Necessary Center Conditions for the Poincaré Systems 𝑃(2,2𝑛)(𝑛≤5)

Abstract

We obtain sufficient and necessary center conditions for the Poincaré system 𝑃(2,2𝑛)(𝑛≤5). The necessity of the condition is derived from the first 2𝑛 focal values by symbolic computation with Maple, and the sufficiency is proved by Volokitin's method.

1. Introduction

Research on Hilbert’s sixteenth problem [1, 2] in general usually proceeds by the investigation on specific classes of polynomial systems. Much effort has been devoted in recent years to the investigation of various systems with cubic or quintic polynomials for the center problem [3–9]. We are interested in a certain family of polynomial systems of the form𝑥̇=−𝑦+𝑥𝑅𝑥,𝑦,𝑦̇=𝑥+𝑦𝑅𝑥,𝑦, (1.1) with 𝑥̇=𝑑𝑥/𝑑𝑡,𝑦̇=𝑑𝑦/𝑑𝑡, and 𝑅(𝑥,𝑦)=∑^𝑛_𝑖=1𝑅_𝑖(𝑥,𝑦), where 𝑅_𝑖 is a homogeneous polynomial of degree 𝑖. The centers of these systems are called uniformly isochronous centers [10]. The case in which 𝑅(𝑥,𝑦) is of degree 𝑖 has been investigated in [10, 11].

In the nonhomogeneous case [12], the pioneering studies mainly focus on the systems with 𝑅(𝑥,𝑦)=𝑅₁+𝑅₂ [13], 𝑅(𝑥,𝑦)=𝑅₁+𝑅₃ [14], 𝑅(𝑥,𝑦)=𝑅₁+𝑅₂+𝑅₃(𝑅²₁+𝑅²₂+𝑅²₃≠0) [15], 𝑅(𝑥,𝑦)=𝑅₁+𝑅₂+𝑅₃+𝑅₄ (𝑅₄≠0 and only one 𝑅_𝑖 not equal to zero, for 𝑖=1,2,3) [16].

In this paper, we consider the system (1.1) with 𝑅(𝑥,𝑦)=𝑅₂+𝑅_2𝑛(𝑛>1), where𝑅₂=𝑎_2,0𝑥²+𝑎_1,1𝑥𝑦+𝑎_0,2𝑦²,𝑅_2𝑛=^2𝑛_𝑖=0𝑏_{2𝑛−𝑖,𝑖}𝑥^2𝑛−𝑖𝑦^𝑖, (1.2) and 𝑏_2,0,𝑏_1,1,𝑏_0,2,𝑏_{2𝑛−𝑖,𝑖}(𝑖=0,…,2𝑛) are real constants. We call the systems above as 𝑃(2,2𝑛). In [17, 18], the authors prove that in the system 𝑃(2,2𝑛) the uniformly isochronous centers are time reversible, which was done by imposing the existence of a transversal commutator. However, for any concrete 𝑛, it is difficult to obtain explicit form of the conditions for the origin to be a center for the system 𝑃(2,2𝑛) due to increasing expansion of computation during the management of large expressions. Sufficient and necessary conditions for the origin to be a center are obtained in Volokitin [19, 20] for 𝑛=2 and 3 and in Xu and Lu [21] for 𝑛=4.

In this paper, we shall consider the system 𝑃(2,10) and obtain sufficient and necessary conditions for the origin to be a center. In Section 2, the main result and the basic method are stated, and the necessary and sufficient center conditions are proved in Sections 3 and 4, respectively. In Section 5, sufficient and necessary center conditions for systems 𝑃(2,2𝑛)(𝑛=2,3 and 4) are given in tight form of polynomial systems.

2. Center Condition

In this section, the technique in [22] is adopted, and a recursive formula for focal values is obtained.

For 𝑛=5, system 𝑃(2,2𝑛) takes the form𝑥̇=−𝑦+𝑥𝑅₂𝑥,𝑦+𝑅₁₀𝑥,𝑦,𝑦̇=𝑥+𝑦𝑅₂𝑥,𝑦+𝑅₁₀𝑥,𝑦. (2.1) In polar coordinate system, system (2.1) can be written as𝑟̇=𝑝𝑟³+𝑞𝑟¹¹,𝜃̇=1, (2.2) where 𝑟̇=𝑑𝑟/𝑑𝜃,𝑝and𝑞are the functions of𝜃:𝑝=𝑎_2,0cos²(𝜃)+𝑎_1,1cos(𝜃)sin(𝜃)+𝑎_0,2sin²(𝜃),𝑞=¹⁰_𝑖=0𝑏_{10−𝑖,𝑖}cos^10−𝑖(𝜃)sin^𝑖(𝜃). (2.3) Let 𝑟(𝜃,𝑐) be the solution of (2.2) with 𝑟(0,𝑐)=𝑐. For 𝑐>0 small enough, we write𝑟(𝜃,𝑐)=^∞_𝑚=1𝑎_𝑚(𝜃)𝑐^𝑚, (2.4) where 𝑎₁(0)=1,𝑎_𝑚(0)=0(𝑚>1). Substituting (2.4) into (2.2) and equating the coefficients of 𝑐, we obtain𝑎̇_𝑚=𝑝_{ℎ+𝑗+𝑙=𝑚ℎ,𝑗,𝑙≥1}𝑎_ℎ𝑎_𝑗𝑎_𝑙+𝑞_{𝑠1+𝑠2+⋯+𝑠11𝑠=𝑚1,𝑠2,…,𝑠11≥1}𝑎_𝑠1𝑎_𝑠2⋯𝑎_𝑠11. (2.5) For (2.5), we can easily obtain 𝑎̇₁=0,𝑎̇_2𝑘=0(𝑘=1,2,…). Thus we can deduce𝑎_𝑚=(),1,𝑚=1,0,𝑚=2𝑘𝑘=1,2,…_𝜃0Φ_𝑚𝑑𝜃,𝑚=2𝑘+1(𝑘=1,2,…), (2.6) whereΦ_𝑚=𝑝_{ℎ+𝑗+𝑙=𝑚ℎ,𝑗,𝑙≥1ℎ,𝑗,𝑙𝑖𝑠𝑜𝑑𝑑}𝑎_ℎ𝑎_𝑗𝑎_𝑙+𝑞_{𝑠1+𝑠2+⋯+𝑠11𝑠=𝑚1,𝑠2,…,𝑠11𝑠≥11,𝑠2,…,𝑠11𝑖𝑠𝑜𝑑𝑑}𝑎_𝑠1𝑎_𝑠2⋯𝑎_𝑠11. (2.7) The value of 𝑎_2𝑘+1(2𝜋)(𝑘=1,2,…) is called the 𝑘th focal value. Let 𝐿_𝑘=𝑎_2𝑘+1(𝑘=1,2,…); then the origin is a center for system (2.1) if and only if𝐷_𝑘≜𝐿_𝑘(2𝜋)=𝑎_2𝑘+1(2𝜋)=0,(𝑘=1,2,…). (2.8) In fact, we have𝐿_𝑘=^𝜃₀Ψ_𝑘𝑑𝜃(𝑘=1,2,…), (2.9) whereΨ_𝑘=𝑝_{ℎ+𝑗+𝑙=𝑘−1ℎ,𝑗,𝑙≥0}𝐿_ℎ𝐿_𝑗𝐿_𝑙+𝑞_{𝑠1+𝑠2+…+𝑠11𝑠=𝑘−51,𝑠2,…,𝑠11≥0}𝐿_𝑠1𝐿_𝑠2⋯𝐿_𝑠11, (2.10) and 𝐿₀=𝑎₁=1.

The main result of the present paper is presented in the following theorem.

Theorem 2.1. The origin is a center for system (2.1) if and only if the following conditions are satisfied: 0=𝑎_2,0+𝑎_0,2,0=63𝑏_0,10+3𝑏_6,4+3𝑏_4,6+7𝑏_8,2+7𝑏_2,8+63𝑏_10,0,0=126𝑎_1,1𝑏_0,10+21𝑎_2,0𝑏_3,7+15𝑎_2,0𝑏_5,5+63𝑎_2,0𝑏_1,9−189𝑏_10,0𝑎_1,1+7𝑎_1,1𝑏_2,8−3𝑎_1,1𝑏_6,4−14𝑎_1,1𝑏_8,2+21𝑎_2,0𝑏_7,3+63𝑎_2,0𝑏_9,1,0=−4𝑎²_2,0𝑏_8,2+𝑎²_1,1𝑏_2,8−108𝑎²_2,0𝑏_10,0−4𝑎²_2,0𝑏_2,8+27𝑎²_1,1𝑏_10,0+27𝑎²_1,1𝑏_0,10+𝑎²_1,1𝑏²_8,2−108𝑎_2,0𝑏_0,10+3𝑎_1,1𝑎_2,0𝑏_3,7−3𝑎_1,1𝑎_2,0𝑏_7,3−18𝑎_2,0𝑎_1,1𝑏_9,1+18𝑎_2,0𝑎_1,1𝑏_1,9,0=72𝑎²_2,0𝑎³_1,1𝑏_10,0+2𝑎²_1,1𝑎³_2,0𝑏_7,3+16𝑎⁴_2,0𝑎_1,1𝑏_6,4−18𝑎³_2,0𝑎²_1,1𝑏_3,7+3𝑎⁴_1,1𝑎_2,0𝑏_7,3+8𝑎⁵_2,0𝑏_7,3+96𝑎⁵_2,0𝑏_1,9+9𝑎⁵_1,1𝑏_0,10−𝑎⁵_1,1𝑏_8,2−36𝑎⁵_1,1𝑏_10,0+96𝑎⁵_2,0𝑏_9,1+8𝑎⁵_2,0𝑏_3,7−156𝑎³_2,0𝑎²_1,1𝑏_1,9−36𝑎³_2,0𝑎²_1,1𝑏_9,1−228𝑎²_2,0𝑎³_1,1b_0,10−12𝑎²_2,0𝑎³_1,1𝑏_8,2−4𝑎³_1,1𝑎²_2,0𝑏_6,4+9𝑎_2,0𝑎⁴_1,1𝑏_1,9+64𝑎⁴_2,0𝑎_1,1𝑏_8,2+768𝑎⁴_2,0𝑎_1,1𝑏_0,10+288𝑎⁴_2,0𝑎_1,1𝑏_10,0+27𝑎⁴_1,1𝑎_2,0𝑏_9,1,0=𝑎⁶_1,1𝑏_8,2+45𝑎⁶_1,1𝑏_10,0−3𝑎⁵_1,1𝑎_2,0𝑏_7,3−36𝑎⁵_1,1𝑎_2,0𝑏_9,1−28𝑎²_2,0𝑎⁴_1,1𝑏_0,10+16𝑎²_2,0𝑎⁴_1,1𝑏_8,2+4𝑎⁴_1,1𝑎²_2,0𝑏_6,4−148𝑎²_2,0𝑎⁴_1,1𝑏_10,0+4𝑎³_2,0𝑎³_1,1𝑏_7,3+76𝑎³_2,0𝑎³_1,1𝑏_9,1−28𝑎³_2,0𝑎³_1,1𝑏_1,9+240𝑎⁴_2,0𝑎²_1,1𝑏_10,0+240𝑎⁴_2,0𝑎²_1,1𝑏_0,10−16𝑎⁵_2,0𝑎_1,1𝑏_9,1+16𝑎⁵_2,0𝑎_1,1𝑏_1,9−64𝑎⁶_2,0𝑏_0,10−64𝑎⁶_2,0𝑏_10,0,0=−35b_10,0𝑎⁹_1,1+35𝑎_2,0𝑏_9,1𝑎⁸_1,1−30𝑎⁷_1,1𝑎²_2,0𝑏_8,2+350𝑎⁷_1,1𝑎²_2,0𝑏_10,0−340𝑎⁶_1,1𝑎³_2,0𝑏_9,1+20𝑎⁶_1,1𝑎³_2,0𝑏_7,3+280𝑎⁵_1,1𝑎⁴_2,0𝑏_8,2−8𝑎⁵_1,1𝑎⁴_2,0𝑏_6,4−600𝑎⁵_1,1𝑎⁴_2,0𝑏_10,0+480𝑎⁴_1,1𝑎⁵_2,0𝑏_9,1−176𝑎⁴_1,1𝑎⁵_2,0𝑏_7,3+64𝑎³_1,1𝑎⁶_2,0𝑏_6,4−288𝑎³_1,1𝑎⁶_2,0𝑏_8,2+800𝑎³_1,1𝑎⁶_2,0𝑏_10,0+96𝑎²_1,1𝑎⁷_2,0𝑏_7,3−448𝑎²_1,1𝑎⁷_2,0𝑏_9,1+128𝑎_1,1𝑎⁸_2,0𝑏_8,2−640𝑎_1,1𝑎⁸_2,0𝑏_10,0+128𝑎⁹_2,0𝑏_9,1+128𝑎⁹_2,0𝑏_1,9. (2.11)

3. The Proof of the Necessity

By simple computation in terms of (2.10) with Maple, we obtain Ψ₁=𝑝𝐿³₀,Ψ₂=3𝑝𝐿²₀𝐿₁,Ψ₃=𝑝3𝐿²₀𝐿₂+3𝐿₀𝐿²₁,Ψ₄=𝑝3𝐿²₀𝐿₃+6𝐿₀𝐿₁𝐿₂+𝐿³₁,Ψ₅=𝑝3𝐿²₀𝐿₄+6𝐿₀𝐿₁𝐿₃+3𝐿₀𝐿²₂+3𝐿²₁𝐿₂+𝑞𝐿₀¹¹,Ψ₆=𝑝3𝐿²₀𝐿₅+6𝐿₀𝐿₁𝐿₄+6𝐿₀𝐿₂𝐿₃+3𝐿²₁𝐿₃+3𝐿₁𝐿²₂+11𝑞𝐿₀¹⁰𝐿₁,Ψ₇=𝑝3𝐿²₁𝐿₄+6𝐿₁𝐿₂𝐿₃+3𝐿₀𝐿²₃+𝐿³₂+3𝐿²₀𝐿₆+6𝐿₀𝐿₁𝐿₅+6𝐿₀𝐿₂𝐿₄+𝑞11𝐿₀¹⁰𝐿₂+55𝐿⁹₀𝐿²₁,Ψ₈=𝑝3𝐿²₂𝐿₃+3𝐿²₁𝐿₅+6𝐿₁𝐿₂𝐿₄+6𝐿₀𝐿₃𝐿₄+6𝐿₀𝐿₁𝐿₆+6𝐿₀𝐿₂𝐿₅+3𝐿₁𝐿²₃+3𝐿²₀𝐿₇+𝑞11𝐿₀¹⁰𝐿₃+110𝐿⁹₀𝐿₁𝐿₂+165𝐿⁸₀𝐿³₁,Ψ₉=𝑝6𝐿₀𝐿₂𝐿₆+6𝐿₀𝐿₁𝐿₇+3𝐿²₀𝐿₈+6𝐿₁𝐿₃𝐿₄+6𝐿₁𝐿₂𝐿₅+3𝐿²₂𝐿₄+3𝐿₂𝐿²₃+3𝐿²₁𝐿₆+3𝐿₀𝐿²₄+6𝐿₀𝐿₃𝐿₅+𝑞110𝐿⁹₀𝐿₁𝐿₃+55𝐿⁹₀𝐿²₂+495𝐿⁸₀𝐿²₁𝐿₂+330𝐿⁷₀𝐿⁴₁+11𝐿₀¹⁰𝐿₄,Ψ₁₀=𝑝𝐿²₀𝐿₉+𝐿²₂𝐿₅+2𝐿₁𝐿₂𝐿₆+𝐿²₁𝐿₇+2𝐿₀𝐿₂𝐿₇+4𝐿₀𝐿₃𝐿₆+4𝐿₂𝐿₃𝐿₄+2𝐿₁𝐿²₄+4𝐿₁𝐿₃𝐿₅+2𝐿₀𝐿₁𝐿₈+4𝐿₀𝐿₄𝐿₅+𝐿³₃+𝑞110𝐿⁹₀𝐿₂𝐿₃+110𝐿⁹₀𝐿₁𝐿₄+11𝐿₀¹⁰𝐿₅+495𝐿⁸₀𝐿₁𝐿²₂+495𝐿⁸₀𝐿²₁𝐿₃+1320𝐿⁷₀𝐿³₁𝐿₂+462𝐿⁶₀𝐿⁵₁,Ψ₁₁=𝑝6𝐿₀𝐿₃𝐿₇+6𝐿₀𝐿₂L₈+3𝐿²₀𝐿₁₀+3𝐿₀𝐿²₅+6𝐿₀𝐿₄𝐿₆+3𝐿²₃𝐿₄+3𝐿₂𝐿²₄+6𝐿₂𝐿₃𝐿₅+3𝐿²₂𝐿₆+3𝐿²₁𝐿₈+6𝐿₁𝐿₄𝐿₅+6𝐿₁𝐿₂𝐿₇+6𝐿₁𝐿₃𝐿₆+6𝐿₀𝐿₁𝐿₉+𝑞11𝐿₀¹⁰𝐿₆+110𝐿⁹₀𝐿₁𝐿₅+55𝐿⁹₀𝐿²₃+110𝐿⁹₀𝐿₂𝐿₄+495𝐿⁸₀𝐿²₁𝐿₄+990𝐿⁸₀𝐿₁𝐿₂𝐿₃+165𝐿⁸₀𝐿³₂+1320𝐿⁷₀𝐿³₁𝐿₃+1980𝐿⁷₀𝐿²₁𝐿²₂+2310𝐿⁶₀𝐿⁴₁𝐿₂+462𝐿⁵₀𝐿⁶₁. (3.1)

Substituting (3.1) into (2.9), after a simple calculation, we obtain𝐿₁=12𝑎_2,0(𝜃)(𝜃)−1cossin2𝑎_0,2(𝜃)(𝜃)+1cossin2𝑎_1,1+12𝑎_2,01𝜃−2𝑎_1,1cos²(𝜃)+12𝑎_0,2𝜃, (3.2) and the first focal value 𝐷₁=𝐿₁(2𝜋)=𝑎_2,0𝜋+𝑎_0,2𝜋. From (2.8), we can see that0=𝑎_2,0+𝑎_0,2 (3.3) must be satisfied if the origin is a center for system (2.1).

Following the discussion above, we can compute the first 𝑘th focal values recursively by the following algorithm, which consists of three steps.

Step 1 (Initialization). Let 𝐿₀𝑝(𝜃)=1,𝑘=1,=𝑎_2,0cos²(𝜃)+𝑎_1,1cos(𝜃)sin(𝜃)−𝑎_2,0sin²(𝜃),𝑞(𝜃)=¹⁰_𝑖=0𝑏_{10−𝑖,𝑖}cos^10−𝑖(𝜃)sin^𝑖(𝜃). (3.4)

Step 2 (Computation of 𝑘th focal value). 𝐿_𝑘 can be obtained in terms of (2.10) and (3.1) with Maple. Then the 𝑘th focal value 𝐷_𝑘 can be obtained by simple calculation.

Step 3 (Let 𝑘=𝑘+1, and go to Step 2). Using computer algebra and writing a Maple code applying the algorithm above, we obtain 𝐷_𝑘(),𝐷=0𝑘=1,2,…,4₅=1𝜋12863𝑏_0,10+3𝑏_6,4+63𝑏_10,0+3𝑏_4,6+7𝑏_8,2+7𝑏_2,8,𝐷₆=1𝜋1287𝑎_1,1𝑏_4,6+14𝑎_2,0𝑏_7,3+10𝑎_2,0𝑏_5,5+21𝑏_10,0𝑎_1,1+21𝑎_1,1𝑏_2,8+7𝑎_1,1𝑏_8,2+42𝑎_2,0𝑏_9,1+5𝑎_1,1𝑏_6,4+231𝑎_1,1𝑏_0,10+14𝑎_2,0𝑏_3,7+42𝑎_2,0𝑏_1,9,𝐷₇=5𝜋512132𝑎²_2,0𝑏_10,0+5𝑎²_1,1𝑏_6,4+5𝑎²_1,1𝑏_8,2+20𝑎²_2,0𝑏_6,4+33𝑎²_1,1𝑏_2,8+36𝑎²_2,0𝑏_8,2+20𝑎²_2,0𝑏_4,6+132𝑎_1,1𝑎_2,0𝑏_1,9+9𝑎²_1,1𝑏_4,6+36𝑎_2,0𝑏_9,1𝑎_1,1+20𝑎_2,0𝑎_1,1𝑏_7,3+20𝑎_2,0𝑎_1,1𝑏_5,5+132𝑎²_2,0𝑏_0,10+36𝑎²_2,0𝑏_2,8+429𝑎²_1,1𝑏_0,10+9𝑎²_1,1𝑏_10,0+36𝑎_1,1𝑎_2,0𝑏_3,7,𝐷₈=5𝜋4096792𝑎³_2,0𝑏_9,1+360𝑎³_2,0𝑏_7,3+6435𝑎³_1,1𝑏_0,10+280𝑎³_2,0𝑏_5,5+45𝑎³_1,1𝑏_10,0+99𝑎³_1,1𝑏_4,6+35𝑎³_1,1𝑏_8,2+429𝑎³_1,1𝑏_2,8+360𝑎³_2,0𝑏_3,7+792𝑎³_2,0𝑏_1,9+540𝑎_1,1𝑎²_2,0𝑏_8,2+210𝑎_2,0𝑎²_1,1𝑏_7,3+270𝑎²_1,1𝑎_2,0𝑏_9,1+540𝑎²_2,0𝑎_1,1𝑏_4,6+5148𝑎²_2,0𝑎_1,1𝑏_0,10+2574𝑎_2,0𝑎²_1,1𝑏_1,9+420𝑎²_2,0𝑎_1,1𝑏_6,4+1188²𝑎_2,0𝑏_10,0𝑎_1,1+1188𝑎²_2,0𝑎_1,1𝑏_2,8+270𝑎_2,0𝑎²_1,1𝑏_5,5+594𝑎²_1,1𝑎_2,0𝑏_3,7+45𝑎³_1,1𝑏_6,4,𝐷₉=35𝜋327681120𝑎³_2,0𝑎_1,1𝑏_7,3+1120𝑎³_2,0𝑎_1,1𝑏_5,5+17160𝑎²_2,0𝑎²_1,1𝑏_0,10+1760𝑎³_2,0𝑎_1,1𝑏_3,7+1144𝑎_2,0𝑎³_1,1𝑏_3,7+5720𝑎_2,0𝑎³_1,1𝑏_1,9+1760𝑎³_2,0𝑎_1,1𝑏_9,1+440𝑎_2,0𝑎³_1,1𝑏_5,5+3432𝑎²_2,0𝑎²_1,1𝑏_2,8+840𝑎²_2,0𝑎²_1,1𝑏_8,2+280𝑎³_1,1𝑎_2,0𝑏_9,1+880𝑎⁴_2,0𝑏_2,8+143𝑎⁴_1,1𝑏_4,6+2288𝑎⁴_2,0𝑏_0,10+2288𝑎⁴_2,0𝑏_10,0+35𝑎⁴_1,1𝑏_10,0+55𝑎⁴_1,1𝑏_6,4+1320𝑎²_1,1𝑎²_2,0𝑏_10,0+4576𝑎_1,1𝑎³_2,0𝑏_1,9+1320𝑎²_2,0𝑎²_1,1𝑏_4,6+280𝑎³_1,1𝑎_2,0𝑏_7,3+12155𝑎⁴_1,1𝑏_0,10+840𝑎²_2,0𝑎²_1,1𝑏_6,4+560𝑎⁴_2,0𝑏_6,4+35𝑎⁴_1,1𝑏_8,2+560𝑎⁴_2,0𝑏_4,6+880𝑎⁴_2,0𝑏_8,2+715𝑎⁴_1,1𝑏_2,8,𝐷₁₀=11𝜋32768²7𝑏_2,8+63𝑏_0,10+63𝑏_10,0+3𝑏_6,4+7𝑏_8,2+3𝑏_4,6+7𝜋163843080𝑎³_1,1𝑎²_2,0𝑏_10,0+34320𝑎_1,1𝑎⁴_2,0𝑏_0,10+17160𝑏_2,8𝑎³_1,1𝑎²_2,0+11440𝑎_1,1𝑎⁴_2,0𝑏_2,8+2520𝑎³_1,1𝑎²_2,0𝑏_8,2+5040𝑎²_1,1𝑎³_2,0𝑏_7,3+24310𝑎⁴_1,1𝑎_2,0𝑏_1,9+6160𝑎_1,1𝑎⁴_2,0𝑏_8,2+2016𝑎⁵_2,0𝑏_5,5+2464𝑎⁵_2,0𝑏_3,7+6160𝑎⁴_2,0𝑎_1,1𝑏_4,6+4290𝑎_2,0𝑎⁴_1,1𝑏_3,7+6160𝑎²_1,1𝑎³_2,0𝑏_9,1+2431𝑏_2,8𝑎⁵_1,1+6160𝑎³_2,0𝑎²_1,1𝑏_5,5+11440𝑎_1,1𝑎⁴_2,0𝑏_10,0+143𝑎⁵_1,1𝑏_6,4+429𝑎⁵_1,1𝑏_4,6+630𝑎⁴_1,1𝑎_2,0𝑏_9,1+770𝑎⁴_1,1𝑎_2,0𝑏_7,3+5040𝑎_1,1𝑎⁴_2,0𝑏_6,4+97240𝑎³_1,1𝑎²_2,0𝑏_0,10+46189𝑎⁵_1,1𝑏_0,10+77𝑎⁵_1,1𝑏_8,2+4576𝑎⁵_2,0𝑏_9,1+1430𝑎⁴_1,1𝑎_2,0𝑏_5,5+2464𝑎⁵_2,0𝑏_7,3+11440𝑎³_2,0𝑎²_1,1𝑏_3,7+63𝑎⁵_1,1𝑏_10,0+34320𝑎²_1,1𝑎³_2,0𝑏_1,9+3080𝑎³_1,1𝑎²_2,0𝑏_6,4+5720𝑎³_1,1𝑎²_2,0𝑏_4,6+4576𝑎⁵_2,0𝑏_1,9,𝐷₁₁=3𝜋4096²7𝑏_2,8+63𝑏_10,0+3𝑏_6,4+7𝑏_8,2+63𝑏_0,10+3𝑏_4,6×7𝑎_1,1𝑏_8,2+10𝑏_5,5𝑎_2,0+5𝑎_1,1𝑏_6,4+14𝑏_7,3𝑎_2,0+7𝑎_1,1𝑏_4,6+42𝑏_9,1𝑎_2,0+14𝑏_3,7𝑎_2,0+231𝑎_1,1𝑏_0,10+21𝑏_2,8𝑎_1,1+21𝑎_1,1𝑏_10,0+42𝑎_2,0𝑏_1,9+1𝜋×196608010752𝑎_2,0𝑏_7,3𝑏_3,7+126𝑎_1,1𝑏_10,0𝑏_3,7+580671𝑎_1,1𝑏_0,10𝑏_9,1+113022𝑎_1,1𝑏_7,3𝑏_0,10+23667𝑎_1,1𝑏_9,1𝑏_10,0−61296𝑎_2,0𝑏_10,0𝑏_4,6+663𝑎_1,1𝑏_1,9𝑏_4,6[phone]𝑎⁵_2,0𝑎_1,1𝑏_1,9[phone]𝑎³_2,0𝑎³_1,1𝑏_3,7[phone]𝑎²_2,0𝑎⁴_1,1𝑏_0,10+7046𝑎_2,0𝑏_5,5𝑏_3,7+7620480𝑎⁵_2,0𝑎_1,1𝑏_7,3+9525600𝑎⁴_2,0𝑎²_1,1𝑏_8,2−143332𝑎_2,0𝑏_2,8𝑏_10,0−129948𝑎_2,0𝑏_4,6𝑏_0,10+9100𝑎_1,1𝑏_2,8𝑏_7,3+9879𝑎_1,1𝑏_9,1𝑏_6,4+50388𝑏_0,10𝑎_1,1𝑏_3,7−268736𝑎_2,0𝑏_2,8𝑏_0,10+140𝑎_1,1𝑏_8,2𝑏_3,7+234𝑎_1,1𝑏_6,4𝑏_3,7+9525600𝑎⁴_2,0𝑎²_1,1𝑏_6,4+5012280𝑎_2,0𝑎⁵_1,1𝑏_3,7+437𝑎_1,1𝑏_6,4𝑏_5,5+54145𝑎_1,1𝑏_5,5𝑏_0,10[phone]𝑎⁵_2,0𝑎_1,1𝑏_3,7[phone]𝑎²_1,1𝑎⁴_2,0𝑏_0,10−129168𝑎_2,0𝑏_6,4𝑏_0,10[phone]𝑎⁴_2,0𝑎²_1,1𝑏_4,6+57330𝑎⁶_1,1𝑏_8,2+122850𝑎⁶_1,1𝑏_6,4−12318𝑎_2,0𝑏²_8,2+7862400𝑎⁶_2,0𝑏_0,10+4914𝑎_2,0𝑏²_3,7−3198𝑎_2,0𝑏²_4,6+2645370𝑎⁶_1,1𝑏_2,8−16354𝑎_2,0𝑏²_2,8−11856𝑎_2,0𝑏_6,4𝑏_8,2+15555𝑎_1,1𝑏_9,1𝑏_4,6+7802𝑎_2,0𝑏_7,3𝑏_5,5+3439800𝑎²_2,0𝑎⁴_1,1𝑏_6,4[phone]𝑎²_1,1𝑎⁴_2,0𝑏_2,8+78636𝑏²_9,1𝑎_2,0+39690𝑎⁶_1,1𝑏_10,0+2540160𝑎⁶_2,0𝑏_6,4−881790𝑎_2,0𝑏²_0,10+417690𝑎⁶_1,1𝑏_4,6+4199𝑎_1,1𝑏_1,9𝑏_2,8+329𝑏_8,2𝑎_1,1𝑏_5,5[phone]𝑎⁶_1,1𝑏_0,10−12220𝑎_2,0𝑏_4,6𝑏_8,2+3669120𝑎⁶_2,0𝑏_2,8+63𝑎_1,1𝑏_1,9𝑏_10,0+6350400𝑎³_2,0𝑎³_1,1𝑏_9,1+91𝑎_1,1𝑏_8,2𝑏_1,9+476280𝑎_2,0𝑎⁵_1,1𝑏_9,1−2946𝑎_2,0𝑏²_6,4+195𝑏_6,4𝑎_1,1𝑏_1,9[phone]𝑎²_2,0𝑎⁴_1,1𝑏_2,8−132672𝑎_2,0𝑏_10,0𝑏_8,2+11367𝑎_1,1𝑏_9,1𝑏_8,2+385𝑎_1,1𝑏_10,0𝑏_5,5+43050𝑎_2,0𝑏_9,1𝑏_3,7+1474200𝑎_2,0𝑎⁵_1,1𝑏_5,5−14768𝑎_2,0𝑏_4,6𝑏_2,8+2381400𝑎²_2,0𝑎⁴_1,1𝑏_8,2+21268𝑏_1,9𝑎_2,0𝑏_5,5+7620480a⁵_2,0𝑎_1,1𝑏_5,5−258076𝑎_2,0𝑏_0,10𝑏_8,2+30498𝑏_1,9𝑎_2,0𝑏_3,7+129024𝑎_2,0𝑏_1,9𝑏_9,1+1302𝑏_7,3𝑎_1,1𝑏_8,2+687960𝑎_2,0𝑎⁵_1,1𝑏_7,3+44766𝑎_2,0𝑏_9,1𝑏_7,3+30956𝑏_9,1𝑎_2,0𝑏_5,5+949𝑎_1,1𝑏_4,6𝑏_5,5+3094𝑎_1,1𝑏_3,7𝑏_2,8[phone]𝑎³_2,0𝑎³_1,1𝑏_1,9+7371000𝑎²_2,0𝑎⁴_1,1𝑏_4,6+3913𝑎_1,1𝑏_5,5𝑏_2,8+32214𝑏_1,9𝑎_2,0𝑏_7,3−28672𝑎_2,0𝑏_8,2𝑏_2,8−60516𝑎_2,0𝑏_10,0𝑏_6,4+5838𝑎_2,0𝑏²_7,3+2381400𝑎²_2,0𝑎⁴_1,1𝑏_10,0−1290240𝑎_2,0𝑏_0,10𝑏_10,0+2004𝑎_1,1𝑏_7,3𝑏_10,0+624𝑎_1,1𝑏_3,7𝑏_4,6+88179𝑎_1,1𝑏_1,9𝑏_0,10+3669120𝑎⁶_2,0𝑏_8,2+2538𝑎_1,1𝑏_7,3𝑏_4,6−408450𝑎_2,0𝑏²_10,0+7862400𝑎⁶_2,0𝑏_10,0[phone]𝑎⁴_2,0𝑎²_1,1𝑏_10,0+9172800𝑎³_2,0𝑎³_1,1𝑏_5,5−14404𝑎_2,0𝑏_2,8𝑏_6,4+2560𝑎_2,0𝑏²_5,5+50267𝑏_2,8𝑏_9,1𝑎_1,1. (3.5) Usually Wu’s method [23] is used to do the triangular sets reduction for the focal values, that is, 𝐷₅,…,𝐷₁₁. The method provides a standard algorithm [24] to handle the reduction of polynomials; however, we here just take use the idea of Wu’s method and do the reduction by substitution method due to the speciality of our case. Thus we can obtain the relations (2.11). Therefore, the necessary part of Theorem 2.1 is proved.

4. The Proof of the Sufficiency

Case 1 (𝑎_1,1=0). From (2.11), we have that either(i)𝑎_0,2=𝑎_2,0=0,63𝑏_0,10+3𝑏_6,4+3𝑏_4,6+7𝑏_8,2+7𝑏_2,8+63𝑏_10,0=0, or(ii)𝑎_0,2+𝑎_2,0=0,𝑏_0,10+𝑏_10,0=0,𝑏_1,9+𝑏_9,1=0,𝑏_2,8+𝑏_8,2=0,𝑏_3,7+𝑏_7,3=0,𝑏_4,6+𝑏_6,4=0,𝑏_5,5=0.
For Case 1, system (2.1) takes the form𝑥̇=−𝑦+𝑥𝑅₁₀𝑥,𝑦,𝑦̇=𝑥+𝑦𝑅₁₀𝑥,𝑦. (4.1) The center condition of system (4.1) is ∫₀^2𝜋𝑅₁₀(sin(𝜃),cos(𝜃))𝑑𝜃=0 (see ([10]). That is, 63𝑏_0,10+3𝑏_6,4+3𝑏_4,6+7𝑏_8,2+7𝑏_2,8+63𝑏_10,0=0. (4.2)
For Case 2, system (2.1) takes the form𝑥̇=−𝑦+𝑥𝑇,𝑦̇=𝑥+𝑦𝑇, (4.3) where 𝑇=𝑎_0,2(𝑦²−𝑥²)+𝑏_0,10+(𝑦¹⁰−𝑥¹⁰)+𝑏_1,9(𝑥𝑦⁹−𝑥⁹𝑦)+𝑏_2,8(𝑥²𝑦⁸−𝑥⁸𝑦²)+𝑏_3,7(𝑥³𝑦⁷−𝑥⁷𝑦³)+𝑏_4,6(𝑥⁴𝑦⁶−𝑥⁶𝑦⁴)). For system (4.3), the following linear transformation: =()1𝑥,𝑦𝑢,𝑣212−1212 (4.4) leads to 𝑢̇=−𝑣+𝑢²𝑣𝐽(𝑢,𝑣),𝑣̇=𝑢+𝑢𝑣²𝐽(𝑢,𝑣), (4.5) where 𝐽(𝑢,𝑣)=𝐽_0,0+𝐽_8,0𝑢⁸+𝐽_6,2𝑢⁶𝑣²+𝐽_4,4𝑢⁴𝑣⁴+𝐽_2,6𝑢²𝑣⁶+𝐽_0,8𝑣⁸, 𝐽_0,0=𝑎_0,2,𝐽_8,0=12563𝑏_2,8+𝑏_4,6+5𝑏_0,10+4𝑏_1,9+2𝑏_3,7,𝐽_6,2=125624𝑏_1,9−4𝑏_3,7+60𝑏_0,10+4𝑏_2,8−4𝑏_4,6,𝐽_4,4=1256−14𝑏_2,8+126𝑏_0,10+6𝑏_4,6,𝐽_2,6=12564𝑏_3,7−24𝑏_1,9+60𝑏_0,10−4𝑏_4,6+4𝑏_2,8,𝐽_0,8=1256−4𝑏_1,9+3𝑏_2,8+𝑏_4,6+5𝑏_0,10−2𝑏_3,7. (4.6) Obviously, in this case, system (4.5) satisfies the condition of symmetry principle [25, page 135]; thus the origin is a center for the system.

Case 2 (𝑎_1,1≠0,𝑎_2,0=0). From (2.11), we have 𝑏_10,0=𝑏_8,2=𝑏_6,4=𝑏_4,6=𝑏_2,8=𝑏_0,10=0. (4.7) Thus system (2.1) takes the form 𝑥̇=−𝑦+𝑥²𝑦𝑎_1,1+𝑆,𝑦̇=𝑥+𝑥𝑦²𝑎_1,1+𝑆, (4.8) where 𝑆=𝑏_9,1𝑥⁸+𝑏_7,3x⁶𝑦²+𝑏_5,5𝑥⁴𝑦⁴+𝑏_3,7𝑥²𝑦⁶+𝑏_1,9𝑦⁸. Obviously, system (4.8) satisfies the condition of symmetry principle; thus the origin is a center for the system.

Case 3 (𝑎_2,0≠0,𝑎_1,1≠0). In this case, (2.11) takes the form 𝑎_0,2=−𝑎_2,0,𝑏_4,67=−3𝑏_2,8−21𝑏_0,10−𝑏_6,4−73𝑏_8,2−21𝑏_10,0,𝑏_5,51=−15𝑎_2,0−14𝑎_1,1𝑏_8,2+21𝑎_2,0𝑏_7,3+63𝑎_2,0𝑏_9,1−3𝑎_1,1𝑏_6,4−189𝑎_1,1𝑏_10,0+21𝑎_2,0𝑏_3,7+126𝑎_1,1𝑏_0,10+63𝑎_2,0𝑏_1,9+7𝑎_1,1𝑏_2,8,𝑏_2,81=−−4𝑎²_2,0+𝑎²_1,13𝑎_1,1𝑎_2,0𝑏_3,7−18𝑎_1,1𝑎_2,0𝑏_9,1+𝑎²_1,1b_8,2+27𝑎²_1,1𝑏_10,0−4𝑎²_2,0𝑏_8,2−108𝑎²_2,0𝑏_10,0−3𝑎_1,1𝑎_2,0𝑏_7,3−108𝑎²_2,0𝑏_0,10+27𝑎²_1,1𝑏_0,10+18𝑎_1,1𝑎_2,0𝑏_1,9,𝑏_3,7=12𝑎³_2,0−4𝑎²_2,0+9𝑎²_1,1×3𝑎⁴_1,1𝑎_2,0𝑏_7,3+9𝑎⁵_1,1𝑏_0,10+16𝑎_1,1𝑎⁴_2,0𝑏_6,4−𝑎⁵_1,1𝑏_8,2+96𝑎⁵_2,0𝑏_1,9+768𝑎_1,1𝑎⁴_2,0𝑏_0,10−36𝑎²_1,1𝑎³_2,0𝑏_9,1−228𝑎³_1,1𝑎²_2,0𝑏_0,10+288𝑎_1,1𝑎⁴_2,0𝑏_10,0−4𝑎³_1,1𝑎²_2,0𝑏_6,4+8𝑎⁵_2,0𝑏_7,3−156𝑎²_1,1𝑎³_2,0𝑏_1,9−12𝑎³_1,1𝑎²_2,0𝑏_8,2+2𝑎²_1,1𝑎³_2,0𝑏_7,3+9𝑎⁴_1,1𝑎_2,0𝑏_1,9+27𝑎⁴_1,1𝑎_2,0𝑏_9,1+72𝑎³_1,1𝑎²_2,0𝑏_10,0+64𝑎_1,1𝑎⁴_2,0𝑏_8,2+96𝑎⁵_2,0𝑏_9,1−36𝑎⁵_1,1𝑏_10,0,𝑏_0,10=14𝑎²_2,07𝑎⁴_1,1−60𝑎²_1,1𝑎²_2,0+16𝑎⁴_2,0×−148𝑎⁴_1,1𝑎²_2,0𝑏_10,0+76𝑎³_1,1𝑎³_2,0𝑏_9,1+𝑎⁶_1,1𝑏_8,2+4𝑎⁴_1,1𝑎²_2,0𝑏_6,4+4𝑎³_1,1𝑎³_2,0𝑏_7,3−28𝑎³_1,1𝑎³_2,0𝑏_1,9+16𝑎_1,1𝑎⁵_2,0𝑏_1,9−3𝑎⁵_1,1𝑎_2,0𝑏_7,3+16𝑎⁴_1,1𝑎²_2,0𝑏_8,2−16𝑎_1,1𝑎⁵_2,0𝑏_9,1−36𝑎⁵_1,1𝑎_2,0𝑏_9,1+240𝑎²_1,1𝑎⁴_2,0𝑏_10,0+45𝑎⁶_1,1𝑏_10,0−64𝑎⁶_2,0𝑏_10,0,𝑏_1,9=1128𝑎⁹_2,0−280𝑎⁵_1,1𝑎⁴_2,0𝑏_8,2−20𝑎⁶_1,1𝑎³_2,0𝑏_7,3−480𝑎⁴_1,1𝑎⁵_2,0𝑏_9,1−64𝑎³_1,1𝑎⁶_2,0𝑏_6,4−800𝑎³_1,1𝑎⁶_2,0𝑏_10,0−128𝑎_1,1𝑎⁸_2,0𝑏_8,2−96𝑎²_1,1𝑎⁷_2,0𝑏_7,3+176𝑎⁴_1,1𝑎⁵_2,0𝑏_7,3−35𝑎⁸_1,1𝑎_2,0𝑏_9,1+30𝑎⁷_1,1𝑎²_2,0𝑏_8,2+8𝑎⁵_1,1𝑎⁴_2,0𝑏_6,4−128𝑎⁹_2,0𝑏_9,1+448𝑎²_1,1𝑎⁷_2,0𝑏_9,1+288𝑎³_1,1𝑎⁶_2,0𝑏_8,2+340𝑎⁶_1,1𝑎³_2,0𝑏_9,1+640𝑎_1,1𝑎⁸_2,0𝑏_10,0+600𝑎⁵_1,1𝑎⁴_2,0𝑏_10,0+35𝑎⁹_1,1𝑏_10,0−350𝑎⁷_1,1𝑎²_2,0𝑏_10,0. (4.9) Substituting (4.9) into (2.1), we get 𝑥̇=−𝑦+𝑥𝑎_2,0𝑥²+𝑎_1,1𝑥𝑦−𝑎_2,0𝑦²𝐻,̇𝑎𝑥,𝑦𝑦=𝑥+𝑦_2,0𝑥²+𝑎_1,1𝑥𝑦−𝑎_2,0𝑦²𝐻𝑥,𝑦, (4.10) where 𝐻=1𝑥,𝑦128𝑎⁹_2,0128𝑎⁹_2,0+128𝑏_10,0𝑎⁸_2,0𝑥⁸+−128𝑏_10,0𝑎_1,1𝑎⁷_2,0+128𝑏_9,1𝑎⁸_2,0𝑦𝑥⁷+128𝑎²_1,1𝑏_10,0𝑎⁶_2,0−128𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑏_10,0𝑎⁸_2,0+128𝑎⁸_2,0𝑏_8,2𝑦²𝑥⁶+128𝑎²_1,1𝑏_9,1𝑎⁶_2,0+128𝑏_9,1𝑎⁸_2,0−256𝑏_10,0𝑎_1,1𝑎⁷_2,0−128𝑎³_1,1𝑏_10,0a⁵_2,0+128𝑎⁸_2,0𝑏_7,3−128𝑎⁷_2,0𝑎_1,1𝑏_8,2𝑦³𝑥⁵+−256𝑎_1,1𝑏_9,1𝑎⁷_2,0−128𝑎³_1,1𝑏_9,1𝑎⁵_2,0+384𝑎²_1,1𝑏_10,0𝑎⁶_2,0+128𝑎⁶_2,0𝑎²_1,1𝑏_8,2+128𝑎⁴_1,1𝑏_10,0𝑎⁴_2,0+128𝑎⁸_2,0𝑏_8,2+128𝑏_10,0𝑎⁸_2,0+128𝑎⁸_2,0𝑏_6,4−128𝑎⁷_2,0𝑎_1,1𝑏_7,3𝑦⁴𝑥⁴+−880𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−512𝑎³_1,1𝑏_10,0𝑎⁵_2,0+128𝑏_9,1𝑎⁸_2,0−256𝑎⁷_2,0𝑎_1,1𝑏_8,2+880𝑎⁵_1,1𝑏_10,0𝑎³_2,0+768𝑎⁵_2,0𝑎³_1,1𝑏_8,2+384𝑎²_1,1𝑏_9,1𝑎⁶_2,0+128𝑎⁸_2,0𝑏_7,3−544𝑎⁶_2,0𝑎²_1,1𝑏_7,3−384𝑏_10,0𝑎_1,1𝑎⁷_2,0+256𝑎⁷_2,0𝑎_1,1𝑏_6,4𝑦⁵𝑥³+192𝑎⁶_2,0𝑎²_1,1𝑏_6,4−800𝑎⁵_1,1𝑏_9,1𝑎³_2,0+832𝑎³_1,1𝑏_9,1𝑎⁵_2,0+688𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−384𝑎_1,1𝑏_9,1𝑎⁷_2,0+192𝑎⁷_2,0𝑎_1,1𝑏_7,3+800𝑎⁶_1,1𝑏_10,0𝑎²_2,0−464𝑎⁵_2,0𝑎³_1,1𝑏_7,3+128𝑎⁸_2,0𝑏_8,2+128𝑏_10,0𝑎⁸_2,0−1040𝑎⁴_1,1𝑏_10,0𝑎⁴_2,0+768𝑎²_1,1𝑏_10,0𝑎⁶_2,0−512𝑎⁶_2,0𝑎²_1,1𝑏_8,2𝑦⁶𝑥²+−384𝑎²_1,1𝑏_9,1𝑎⁶_2,0−160𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3+240𝑎³_2,0𝑎⁵_1,1𝑏_8,2+96𝑎⁶_2,0𝑎²_1,1𝑏_7,3+64𝑎⁵_2,0𝑎³_1,1𝑏_6,4−256𝑎⁵_2,0𝑎³_1,1𝑏_8,2−480𝑎⁵_1,1𝑏_10,0𝑎³_2,0−512𝑏_10,0𝑎_1,1𝑎⁷_2,0+128𝑎⁷_2,0𝑎_1,1𝑏_8,2+280𝑎⁷_1,1𝑏_10,0𝑎_2,0+400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−280𝑎⁶_1,1𝑏_9,1𝑎²_2,0+640𝑎³_1,1𝑏_10,0𝑎⁵_2,0+128𝑏_9,1𝑎⁸_2,0𝑦⁷𝑥+16𝑎⁵_2,0𝑎³_1,1𝑏_7,3−80𝑎³_1,1𝑏_9,1𝑎⁵_2,0+60𝑎⁵_1,1𝑏_9,1𝑎³_2,0+120𝑎⁴_1,1𝑏_10,0𝑎⁴_2,0−20𝑎³_2,0𝑎⁵_1,1𝑏_7,3−70𝑎⁶_1,1𝑏_10,0𝑎²_2,0+30𝑎²_2,0𝑎⁶_1,1𝑏_8,2+32𝑎⁶_2,0𝑎²_1,1𝑏_8,2−160𝑎²_1,1𝑏_10,0𝑎⁶_2,0−35𝑎⁷_1,1𝑏_9,1𝑎_2,0+8𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4+128𝑏_10,0𝑎⁸_2,0+64𝑎_1,1𝑏_9,1𝑎⁷_2,0−40𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+35𝑎⁸_1,1𝑏_10,0𝑦⁸. (4.11)
For system (4.10), we consider the following linear transformation:x,𝑦=(𝑢,𝑣),𝑡−11𝑡 (4.12) where the diagonal elements of the transformation matrix satisfy the equation 𝑎_2,0𝑡²−𝑎_1,1𝑡−𝑎_2.0=0. Applying the transformation (4.12) to (4.10), we obtain 𝑢̇=−𝑣+𝑢²𝑣𝑀⋅𝐼(𝑢,𝑣),𝑣̇=𝑢+𝑢𝑣²𝑀⋅𝐼(𝑢,𝑣), (4.13) where 𝑀=(4𝑎_2,0𝑡−𝑎_1,1+𝑎_1,1𝑡²)/128𝑎⁹_2,0,𝐼(𝑢,𝑣)=𝐼_0,0+𝐼_8,0𝑢⁸+𝐼_6,2𝑢⁶𝑣²+𝐼_4,4𝑢⁴𝑣⁴+𝐼_2,6𝑢²𝑣⁶+𝐼_0,8𝑣⁸, and 𝐼_0,0,𝐼_8,0,𝐼_6,2,𝐼_4,4,𝐼_2,6,𝐼_0,8 are given in the appendix.
Obviously, system (4.13) satisfies the conditions of symmetry principle, thus the origin is a center for the system. Therefore, the sufficient part of Theorem 2.1 is proved.

5. Center Conditions of 𝑃(2,2𝑛)for𝑛=2,3and4

By similar steps discussed above, we can obtain the following theorem.

Theorem 5.1. The origin is a center for system 𝑃(2,8) if and only if the following conditions are satisfied: 0=𝑎_2,0+𝑎_0,2,0=5𝑏_6,2+35𝑏_0,8+5𝑏_2,6+3𝑏_4,4+35𝑏_8,0,0=𝑎_1,1𝑏_2,6+3𝑎_2,0𝑏_5,3+14𝑎_1,1𝑏_0,8+7𝑎_2,0𝑏_1,7+3𝑎_2,0𝑏_3,5−𝑎_1,1𝑏_6,2+7𝑎_2,0𝑏_7,1−14𝑎_1,1𝑏_8,0,0=7𝑎_2,0𝑏_1,7𝑎_1,1−14𝑎_2,0𝑎_1,1𝑏_7,1−3𝑎_2,0𝑎_1,1𝑏_5,3+𝑎²_1,1𝑏_6,2+21𝑏_8,0𝑎²_1,1+7𝑎²_1,1𝑏_0,8−2𝑎²_2,0𝑏_6,2−56𝑎²_2,0𝑏_0,8−2𝑎²_2,0𝑏_2,6−56𝑎²_2,0𝑏_8,0,0=−3𝑎²_1,1𝑎_2,0𝑏_5,3−48𝑎_1,1𝑎²_2,0𝑏_8,0+18𝑎³_1,1b_8,0−11𝑎²_1,1𝑎_2,0𝑏_7,1+𝑎³_1,1𝑏_6,2−64𝑎²_2,0𝑎_1,1𝑏_0,8−4𝑎_1,1𝑎²_2,0𝑏_6,2+10𝑎³_1,1𝑏_0,8+10𝑎²_1,1𝑎_2,0𝑏_1,7−4𝑎³_2,0𝑏_1,7−4𝑎³_2,0𝑏_7,1,0=8𝑎³_2,0𝑎³_1,1𝑏_7,1+16𝑎⁴_2,0𝑎²_1,1𝑏_8,0−5𝑎⁵_1,1𝑎_2,0𝑏_7,1+5𝑎⁶_1,1𝑏_8,0−16𝑎⁶_2,0𝑏_8,0−16𝑎⁶_2,0𝑏_0,8+4a²_2,0𝑎⁴_1,1𝑏_6,2−2𝑎³_2,0𝑎³_1,1𝑏_5,3−8𝑎⁵_2,0𝑎_1,1𝑏_7,1−4𝑎⁴_2,0𝑎²_1,1𝑏_6,2−10𝑎²_2,0𝑎⁴_1,1𝑏_8,0. (5.1)

The relations above give a complete set of relations in the sense given in [21], that is, this set of relations cover all the cases of the conditions which are sufficient and necessary for the system to be of center type.

Theorem 5.2. The origin is a center for system 𝑃(2,6) if and only if the following conditions are satisfied: 0=𝑎_2,0+𝑎_0,2,0=5𝑏_0,6+𝑏_2,4+𝑏_4,2+5𝑏_6,0,0=3𝑎_2,0𝑏_3,3−𝑎_1,1𝑏_4,2+5𝑎_1,1𝑏_0,6+5𝑎_2,0𝑏_1,5+5𝑎_2,0𝑏_5,1−10𝑎_1,1𝑏_6,0,0=𝑎²_1,1𝑏_0,6+𝑏_6,0𝑎²_1,1−4𝑎²_2,0𝑏_0,6−4𝑎²_2,0𝑏_6,0+𝑎_2,0𝑎_1,1𝑏_1,5−𝑎_2,0𝑎_1,1𝑏_5,1,0=18𝑎²_2,0𝑎³_1,1𝑏_6,0−16𝑎³_2,0𝑎²_1,1𝑏_5,1−24𝑎⁴_2,0𝑎_1,1𝑏_6,0−3𝑎⁵_1,1𝑏_6,0+3𝑎⁴_1,1𝑎_2,0𝑏_5,1+8𝑎⁵_2,0𝑏_1,5+8𝑎⁵_2,0𝑏_5,1−2𝑎³_1,1𝑎²_2,0𝑏_4,2+8𝑎_1,1𝑎⁴_2,0𝑏_4,2. (5.2)

The relations above is another form as the result of paper [20].

Theorem 5.3. The origin is a center for system 𝑃(2,4) if and only if the following conditions are satisfied: 0=𝑎_2,0+𝑎_0,2,0=3𝑏_0,4+𝑏_2,2+3𝑏_4,0,0=𝑎_1,1𝑏_0,4+𝑎_2,0𝑏_1,3+𝑎_2,0𝑏_3,1−𝑎_1,1𝑏_4,0,0=−2𝑎²_2,0𝑏_0,4−𝑎_1,1𝑎_2,0𝑏_3,1+𝑎²_1,1𝑏_4,0−2𝑎²_2,0𝑏_4,0. (5.3)

The relations above is another form of center condition for the system 𝑃(2,4), which has been discussed by Volokitin in paper [19].

Appendix

𝐼_0,0,𝐼_8,0,𝐼_6,2,𝐼_4,4,𝐼_2,6,𝐼_0,8

𝐼_0,0=128𝑎⁹_2,0,𝐼_8,0=128𝑏_10,0𝑎⁸_2,0𝑡⁸+128𝑎⁷_2,0𝑏_10,0𝑎_1,1−128𝑎⁸_2,0𝑏_9,1𝑡⁷+128𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−128𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2𝑡⁶+256𝑎⁷_2,0𝑏_10,0𝑎_1,1−128𝑎⁸_2,0𝑏_9,1+128𝑎⁵_2,0𝑎³_1,1𝑏_10,0−128𝑎⁶_2,0𝑎²_1,1𝑏_9,1−128𝑎⁸_2,0𝑏_7,3+128𝑎⁷_2,0𝑎_1,1𝑏_8,2𝑡⁵+384𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−256𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−128𝑎⁷_2,0𝑎_1,1𝑏_7,3+128𝑎⁸_2,0𝑏_8,2+128𝑎⁶_2,0𝑎²_1,1𝑏_8,2+128𝑎⁸_2,0𝑏_6,4−128𝑎⁵_2,0𝑎³_1,1𝑏_9,1𝑡⁴+−256𝑎⁷_2,0𝑎_1,1𝑏_6,4−880𝑎⁵_1,1𝑏_10,0𝑎³_2,0−128𝑎⁸_2,0𝑏_7,3+512𝑎⁵_2,0𝑎³_1,1𝑏_10,0−384𝑎⁶_2,0𝑎²_1,1𝑏_9,1−768𝑎⁵_2,0𝑎³_1,1𝑏_8,2−128𝑎⁸_2,0𝑏_9,1+256𝑎⁷_2,0𝑎_1,1𝑏_8,2+384𝑎⁷_2,0𝑏_10,0𝑎_1,1+880𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+544𝑎⁶_2,0𝑎²_1,1𝑏_7,3𝑡³+800𝑎²_2,0𝑎⁶_1,1𝑏_10,0+128𝑎⁸_2,0𝑏_8,2−800𝑎³_2,0𝑎⁵_1,1𝑏_9,1−512𝑎⁶_2,0𝑎²_1,1𝑏_8,2+192𝑎⁷_2,0𝑎_1,1𝑏_7,3+832𝑎⁵_2,0𝑎³_1,1𝑏_9,1+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−464𝑎⁵_2,0𝑎³_1,1𝑏_7,3−384𝑎_1,1𝑏_9,1𝑎⁷_2,0+688𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+192𝑎⁶_2,0𝑎²_1,1𝑏_6,4−1040𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0𝑡²+−128𝑎⁷_2,0𝑎_1,1𝑏_8,2−64𝑎⁵_2,0𝑎³_1,1𝑏_6,4−640𝑎⁵_2,0𝑎³_1,1𝑏_10,0−280𝑎⁷_1,1𝑏_10,0𝑎_2,0−128𝑎⁸_2,0𝑏_9,1+512𝑎⁷_2,0𝑏_10,0𝑎_1,1−240𝑎³_2,0𝑎⁵_1,1𝑏_8,2+160𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+480𝑎⁵_1,1𝑏_10,0𝑎³_2,0−96𝑎⁶_2,0𝑎²_1,1𝑏_7,3+384𝑎⁶_2,0𝑎²_1,1𝑏_9,1+256𝑎⁵_2,0𝑎³_1,1𝑏_8,2+280𝑎⁶_1,1𝑏_9,1𝑎²_2,0𝑡+30𝑎²_2,0𝑎⁶_1,1𝑏_8,2−20𝑎³_2,0𝑎⁵_1,1𝑏_7,3+128𝑏_10,0𝑎⁸_2,0+8𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4−70𝑎²_2,0𝑎⁶_1,1𝑏_10,0−35𝑎⁷_1,1𝑏_9,1𝑎_2,0−40𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+60𝑎³_2,0𝑎⁵_1,1𝑏_9,1−80𝑎⁵_2,0𝑎³_1,1𝑏_9,1+120𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+16𝑎⁵_2,0𝑎³_1,1𝑏_7,3+64𝑎_1,1𝑏_9,1𝑎⁷_2,0−160𝑎⁶_2,0𝑎²_1,1𝑏_10,0+35𝑎⁸_1,1𝑏_10,0+32𝑎⁶_2,0𝑎²_1,1𝑏_8,2, (A.1)𝐼_6,2=128𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−128𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2𝑡⁸+−384𝑎⁸_2,0𝑏_7,3+384𝑎⁵_2,0𝑎³_1,1𝑏_10,0+384𝑎⁷_2,0𝑎_1,1𝑏_8,2−128𝑎⁷_2,0𝑏_10,0𝑎_1,1−384𝑎⁶_2,0𝑎²_1,1𝑏_9,1+512𝑎⁸_2,0𝑏_9,1𝑡⁷+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+768𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−768𝑎⁵_2,0𝑎³_1,1𝑏_9,1+768𝑎⁸_2,0𝑏_6,4+2816𝑏_10,0𝑎⁸_2,0+768𝑎⁶_2,0𝑎²_1,1𝑏_8,2−768𝑎⁸_2,0𝑏_8,2−768𝑎⁷_2,0𝑎_1,1𝑏_7,3𝑡⁶+−1920𝑎⁶_2,0𝑎²_1,1𝑏_9,1−2048𝑎⁸_2,0𝑏_9,1+5440𝑎⁶_2,0𝑎²_1,1𝑏_7,3+640𝑎⁷_2,0𝑎_1,1𝑏_8,2−8800𝑎⁵_1,1𝑏_10,0𝑎³_2,0+2688𝑎⁷_2,0𝑏_10,0𝑎_1,1−7680𝑎⁵_2,0𝑎³_1,1𝑏_8,2+8800𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−2560𝑎⁷_2,0𝑎_1,1𝑏_6,4+640𝑎⁸_2,0𝑏_7,3+3200𝑎⁵_2,0𝑎³_1,1𝑏_10,0𝑡⁵+14528𝑎⁵_2,0𝑎³_1,1𝑏_9,1−12000𝑎³_2,0𝑎⁵_1,1𝑏_9,1−9728𝑎⁶_2,0𝑎²_1,1𝑏_8,2−2048𝑎⁸_2,0𝑏_6,4+12000𝑎²_2,0𝑎⁶_1,1𝑏_10,0+7296𝑎⁶_2,0𝑎²_1,1𝑏_10,0+4928𝑎⁷_2,0𝑎_1,1𝑏_7,3+1792𝑏_10,0𝑎⁸_2,0+10320𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−17648𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−3584𝑎_1,1𝑏_9,1𝑎⁷_2,0+1792𝑎⁸_2,0𝑏_8,2−6960𝑎⁵_2,0𝑎³_1,1𝑏_7,3+2880𝑎⁶_2,0𝑎²_1,1𝑏_6,4𝑡⁴+−1344𝑎⁵_2,0𝑎³_1,1𝑏_6,4+640𝑎⁸_2,0𝑏_7,3−19840𝑎⁵_2,0𝑎³_1,1𝑏_10,0+3840𝑎⁷_2,0𝑎_1,1𝑏_6,4−21600𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+12544𝑎⁶_2,0𝑎²_1,1𝑏_9,1−5248𝑎⁷_2,0𝑎_1,1𝑏_8,2+3360𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3+16896𝑎⁵_2,0𝑎³_1,1𝑏_8,2−2048𝑎⁸_2,0𝑏_9,1−5880𝑎⁷_1,1𝑏_10,0𝑎_2,0−10176𝑎⁶_2,0𝑎²_1,1𝑏_7,3+23280𝑎⁵_1,1𝑏_10,0𝑎³_2,0+7552𝑎⁷_2,0𝑏_10,0𝑎_1,1+5880𝑎⁶_1,1𝑏_9,1𝑎²_2,0−5040𝑎³_2,0𝑎⁵_1,1𝑏_8,2𝑡³+−11560𝑎²_2,0𝑎⁶_1,1𝑏_10,0+840𝑎²_2,0𝑎⁶_1,1𝑏_8,2+980𝑎⁸_1,1𝑏_10,0+2816𝑏_10,0𝑎⁸_2,0−560𝑎³_2,0𝑎⁵_1,1𝑏_7,3+7808𝑎⁶_2,0𝑎²_1,1𝑏_8,2+16608𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+6016𝑎⁵_2,0𝑎³_1,1𝑏_7,3+4864𝑎_1,1𝑏_9,1𝑎⁷_2,0+224𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4−2304𝑎⁶_2,0𝑎²_1,1𝑏_6,4−12992𝑎⁵_2,0𝑎³_1,1𝑏_9,1+768𝑎⁸_2,0𝑏_6,4−11392𝑎⁶_2,0𝑎²_1,1𝑏_10,0+11280𝑎³_2,0𝑎⁵_1,1𝑏_9,1−9376𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−980𝑎⁷_1,1𝑏_9,1𝑎_2,0−768𝑎⁸_2,0𝑏_8,2−3072𝑎⁷_2,0𝑎_1,1𝑏_7,3𝑡²+5440𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+448𝑎⁵_2,0𝑎³_1,1𝑏_6,4+1680𝑎³_2,0𝑎⁵_1,1𝑏_8,2−1120𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3+6016𝑎⁵_2,0𝑎³_1,1𝑏_10,0−3840𝑎⁶_2,0𝑎²_1,1𝑏_9,1−6000𝑎⁵_1,1𝑏_10,0𝑎³_2,0+512𝑎⁸_2,0𝑏_9,1−384𝑎⁸_2,0𝑏_7,3+1960𝑎⁷_1,1𝑏_10,0𝑎_2,0−4096𝑎⁵_2,0𝑎³_1,1𝑏_8,2−1960𝑎⁶_1,1𝑏_9,1𝑎²_2,0−768𝑎⁷_2,0𝑎_1,1𝑏_6,4−2432𝑎⁷_2,0𝑏_10,0𝑎_1,1+2304𝑎⁶_2,0𝑎²_1,1𝑏_7,3+1664𝑎⁷_2,0𝑎_1,1𝑏_8,2𝑡+800𝑎²_2,0𝑎⁶_1,1𝑏_10,0+128𝑎⁸_2,0𝑏_8,2−800𝑎³_2,0𝑎⁵_1,1𝑏_9,1−512𝑎⁶_2,0𝑎²_1,1𝑏_8,2+192𝑎⁷_2,0𝑎_1,1𝑏_7,3+832𝑎⁵_2,0𝑎³_1,1𝑏_9,1+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−464𝑎⁵_2,0𝑎³_1,1𝑏_7,3−384𝑎_1,1𝑏_9,1𝑎⁷_2,0+688𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+192𝑎⁶_2,0𝑎²_1,1𝑏_6,4−1040𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0, (A.2)𝐼_4,4=128𝑏_10,0𝑎⁸_2,0+128𝑎⁸_2,0𝑏_6,4−128𝑎⁷_2,0𝑎_1,1𝑏_7,3+128𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+128𝑎⁶_2,0𝑎²_1,1𝑏_8,2+384𝑎⁶_2,0𝑎²_1,1𝑏_10,0−256𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2−128𝑎⁵_2,0𝑎³_1,1𝑏_9,1𝑡⁸+640𝑎⁷_2,0𝑎_1,1𝑏_8,2−1280𝑎⁶_2,0𝑎²_1,1𝑏_9,1−3840𝑎⁵_2,0𝑎³_1,1𝑏_8,2−1280𝑎⁷_2,0𝑎_1,1𝑏_6,4−4400𝑎⁵_1,1𝑏_10,0𝑎³_2,0+1920𝑎⁵_2,0𝑎³_1,1𝑏_10,0+640𝑎⁷_2,0𝑏_10,0𝑎_1,1+4400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+2720𝑎⁶_2,0𝑎²_1,1𝑏_7,3𝑡⁷+14528𝑎⁵_2,0𝑎³_1,1𝑏_9,1−12000𝑎³_2,0𝑎⁵_1,1𝑏_9,1−9728𝑎⁶_2,0𝑎²_1,1𝑏_8,2+12000𝑎²_2,0𝑎⁶_1,1𝑏_10,0−2048𝑎⁸_2,0𝑏_6,4+7296𝑎⁶_2,0𝑎²_1,1𝑏_10,0+4928𝑎⁷_2,0𝑎_1,1𝑏_7,3+10320𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+1792𝑏_10,0𝑎⁸_2,0−17648𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−3584𝑎_1,1𝑏_9,1𝑎⁷_2,0−6960𝑎⁵_2,0𝑎³_1,1𝑏_7,3+1792𝑎⁸_2,0𝑏_8,2+2880𝑎²_1,1𝑏_6,4𝑡⁶+9800𝑎⁶_1,1𝑏_9,1𝑎²_2,0−9800𝑎⁷_1,1𝑏_10,0𝑎_2,0−19680𝑎⁶_2,0𝑎²_1,1𝑏_7,3+9600𝑎⁷_2,0𝑏_10,0𝑎_1,1+21120𝑎⁶_2,0𝑎²_1,1𝑏_9,1+43200𝑎⁵_1,1𝑏_10,0𝑎³_2,0−33920𝑎⁵_2,0𝑎³_1,1𝑏_10,0−8400𝑎³_2,0𝑎⁵_1,1𝑏_8,2+5600𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−8320𝑎⁷_2,0𝑎_1,1𝑏_8,2−40400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+32000𝑎⁵_2,0𝑎³_1,1𝑏_8,2−2240𝑎⁵_2,0𝑎³_1,1𝑏_6,4+7680𝑎⁷_2,0𝑎_1,1𝑏_6,4𝑡⁵+−43488𝑎^52,0𝑎³_1,1𝑏_9,1+19680𝑎⁵_2,0𝑎³_1,1𝑏_7,3+4608𝑎⁸_2,0𝑏_6,4+27328𝑎⁶_2,0𝑎²_1,1𝑏_8,2+560𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4−1400𝑎³_2,0𝑎⁵_1,1𝑏_7,3+12288𝑏_10,0𝑎⁸_2,0−5632𝑎⁸_2,0𝑏_8,2+2450𝑎⁸_1,1𝑏_10,0−36900𝑎²_2,0𝑎⁶_1,1𝑏_10,0−12288𝑎⁷_2,0𝑎_1,1𝑏_7,3+15744𝑎_1,1𝑏_9,1𝑎⁷_2,0−2450𝑎⁷_1,1𝑏_9,1𝑎_2,0−7680𝑎⁶_2,0𝑎²_1,1𝑏_6,4−33216𝑎⁶_2,0𝑎²_1,1𝑏_10,0+54608𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+36200𝑎³_2,0𝑎⁵_1,1𝑏_9,1−30320𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+2100𝑎²_2,0𝑎⁶_1,1𝑏_8,2𝑡⁴+19680𝑎⁶_2,0𝑎²_1,1𝑏_7,3+2240𝑎⁵_2,0𝑎³_1,1𝑏_6,4+9800𝑎⁷_1,1𝑏_10,0𝑎_2,0−5600𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−9600𝑎⁷_2,0𝑏_10,0𝑎_1,1+8320𝑎⁷_2,0𝑎_1,1𝑏_8,2−9800𝑎⁶_1,1𝑏_9,1𝑎²_2,0+8400𝑎³_2,0𝑎⁵_1,1𝑏_8,2−43200𝑎⁵_1,1𝑏_10,0𝑎³_2,0−7680𝑎⁷_2,0𝑎_1,1𝑏_6,4+33920𝑎⁵_2,0𝑎³_1,1𝑏_10,0+40400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−21120𝑎⁶_2,0𝑎²_1,1𝑏_9,1−32000𝑎⁵_2,0𝑎_1,1𝑏_8,2𝑡³+14528𝑎⁵_2,0𝑎³_1,1𝑏_9,1−12000𝑎³_2,0𝑎⁵_1,1𝑏_9,1−9728𝑎⁶_2,0𝑎²_1,1𝑏_8,2+12000𝑎²_2,0𝑎⁶_1,1𝑏_10,0−2048𝑎⁸_2,0𝑏_6,4+7296𝑎⁶_2,0𝑎²_1,1𝑏_10,0+4928𝑎⁷_2,0𝑎_1,1𝑏_7,3+10320𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+1792𝑏_10,0𝑎⁸_2,0−17648𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−3584𝑎_1,1𝑏_9,1𝑎⁷_2,0−6960𝑎⁵_2,0𝑎³_1,1𝑏_7,3+1792𝑎⁸_2,0𝑏_8,2+2880𝑎⁶_2,0𝑎²_1,1𝑏_6,4𝑡²+3840𝑎⁵_2,0𝑎³_1,1𝑏_8,2+1280𝑎⁷_2,0𝑎_1,1𝑏_6,4+1280𝑎⁶_2,0𝑎²_1,1𝑏_9,1−640𝑎⁷_2,0𝑎_1,1𝑏_8,2+4400𝑎⁵_1,1𝑏_10,0𝑎³_2,0−640𝑎⁷_2,0𝑏_10,0𝑎_1,1−2720𝑎⁶_2,0𝑎²_1,1𝑏_7,3−4400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−1920𝑎⁵_2,0𝑎³_1,1𝑏_10,0𝑡+128𝑏_10,0𝑎⁸_2,0+128𝑎⁸_2,0𝑏_6,4−128𝑎⁷_2,0𝑎_1,1𝑏_7,3+128𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+128𝑎⁶_2,0𝑎²_1,1𝑏_8,2+384𝑎⁶_2,0𝑎²_1,1𝑏_10,0−256𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2−128𝑎⁵_2,0𝑎³_1,1𝑏_9,1, (A.3)𝐼_2,6=128𝑎⁸_2,0𝑏_8,2+128𝑏_10,0𝑎⁸_2,0+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+832𝑎⁵_2,0𝑎³_1,1𝑏_9,1−464𝑎⁵_2,0𝑎³_1,1𝑏_7,3+800𝑎²_2,0𝑎⁶_1,1𝑏_10,0−512𝑎⁶_2,0𝑎²_1,1𝑏_8,2+688𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−1040𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−800𝑎³_2,0𝑎⁵_1,1𝑏_9,1+192𝑎⁶_2,0𝑎²_1,1𝑏_6,4+192𝑎⁷_2,0𝑎_1,1𝑏_7,3−384𝑎_1,1𝑏_9,1𝑎⁷_2,0𝑡⁸+3840𝑎⁶_2,0𝑎²_1,1𝑏_9,1−6016𝑎⁵_2,0𝑎³_1,1𝑏_10,0−512𝑎⁸_2,0𝑏_9,1+4096𝑎⁵_2,0𝑎³_1,1𝑏_8,2−448𝑎⁵_2,0𝑎³_1,1𝑏_6,4+1120𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−5440𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+384𝑎⁸_2,0𝑏_7,3+2432𝑎⁷_2,0𝑏_10,0𝑎_1,1−1664𝑎⁷_2,0𝑎_1,1𝑏_8,2−1680𝑎³_2,0𝑎⁵_1,1𝑏_8,2−2304𝑎⁶_2,0𝑎²_1,1𝑏_7,3+1960𝑎⁶_1,1𝑏_9,1𝑎²_2,0−1960𝑎⁷_1,1𝑏_10,0𝑎_2,0+768𝑎⁷_2,0𝑎_1,1𝑏_6,4+6000𝑎⁵_1,1𝑏_10,0𝑎³_2,0𝑡⁷+768𝑎⁸_2,0𝑏_6,4−768𝑎⁸_2,0𝑏_8,2+11280𝑎³_2,0𝑎⁵_1,1𝑏_9,1−11392𝑎⁶_2,0𝑎²_1,1𝑏_10,0−980𝑎⁷_1,1𝑏_9,1𝑎_2,0+16608𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−3072𝑎⁷_2,0𝑎_1,1𝑏_7,3−2304𝑎⁶_2,0𝑎²_1,1𝑏_6,4+4864𝑎_1,1𝑏_9,1𝑎⁷_2,0−12992𝑎⁵_2,0𝑎³_1,1𝑏_9,1−560𝑎³_2,0𝑎⁵_1,1𝑏_7,3+980𝑎⁸_1,1𝑏_10,0+224𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4+6016𝑎⁵_2,0𝑎³_1,1𝑏_7,3+2816𝑏_10,0𝑎⁸_2,0+7808𝑎⁶_2,0𝑎²_1,1𝑏_8,2+840𝑎²_2,0𝑎⁶_1,1𝑏_8,2−11560𝑎²_2,0𝑎⁶_1,1𝑏_10,0−9376𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2𝑡⁶+5248𝑎⁷_2,0𝑎_1,1𝑏_8,2+21600𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−7552𝑎⁷_2,0𝑏_10,0𝑎_1,1+1344𝑎⁵_2,0𝑎³_1,1𝑏_6,4−3840𝑎⁷_2,0𝑎_1,1𝑏_6,4−23280𝑎⁵_1,1𝑏_10,0𝑎³_2,0−640𝑎⁸_2,0𝑏_7,3+10176𝑎⁶_2,0𝑎²_1,1𝑏_7,3−5880𝑎⁶_1,1𝑏_9,1𝑎²_2,0−12544𝑎⁶_2,0𝑎²_1,1𝑏_9,1−3360𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−16896𝑎⁵_2,0𝑎³_1,1𝑏_8,2+5040𝑎³_2,0𝑎⁵_1,1𝑏_8,2+19840𝑎⁵_2,0𝑎³_1,1𝑏_10,0+5880𝑎⁷_1,1𝑏_10,0𝑎_2,0+2048𝑎⁸_2,0𝑏_9,1𝑡⁵+−2048𝑎⁸_2,0𝑏_6,4−6960𝑎⁵_2,0𝑎³_1,1𝑏_7,3−12000𝑎³_2,0𝑎⁵_1,1𝑏_9,1+7296𝑎⁶_2,0𝑎²_1,1𝑏_10,0−9728𝑎⁶_2,0𝑎²_1,1𝑏_8,2+1792𝑎⁸_2,0𝑏_8,2+12000𝑎²_2,0𝑎⁶_1,1𝑏_10,0+14528𝑎⁵_2,0𝑎³_1,1𝑏_9,1+2880𝑎⁶_2,0𝑎²_1,1𝑏_6,4−3584𝑎_1,1𝑏_9,1𝑎⁷_2,0+1792𝑏_10,0𝑎⁸_2,0+10320𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−17648𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+4928𝑎⁷_2,0𝑎_1,1𝑏_7,3𝑡⁴+−640𝑎⁷_2,0𝑎_1,1𝑏_8,2+2048𝑎⁸_2,0𝑏_9,1−640𝑎⁸_2,0𝑏_7,3−5440𝑎⁶_2,0𝑎²_1,1𝑏_7,3−3200𝑎⁵_2,0𝑎³_1,1𝑏_10,0−8800𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0+7680𝑎⁵_2,0𝑎³_1,1𝑏_8,2−2688𝑎⁷_2,0𝑏_10,0𝑎_1,1+1920𝑎⁶_2,0𝑎²_1,1𝑏_9,1+8800𝑎⁵_1,1𝑏_10,0𝑎³_2,0+2560𝑎⁷_2,0𝑎_1,1𝑏_6,4𝑡³+−768𝑎⁵_2,0𝑎³_1,1𝑏_9,1−768𝑎⁸_2,0𝑏_8,2+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+2816𝑏_10,0𝑎⁸_2,0+768𝑎⁸_2,0𝑏_6,4+768𝑎⁶_2,0𝑎²_1,1𝑏_8,2−768𝑎⁷_2,0𝑎_1,1𝑏_7,3+768𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0𝑡²+384𝑎⁸_2,0𝑏_7,3−384𝑎⁷_2,0𝑎_1,1𝑏_8,2−384𝑎⁵_2,0𝑎³_1,1𝑏_10,0+384𝑎⁶_2,0𝑎²_1,1𝑏_9,1−512𝑎⁸_2,0𝑏_9,1+128𝑎⁷_2,0𝑏_10,0𝑎_1,1𝑡+128𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−128𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2, (A.4)𝐼_0,8=16𝑎⁵_2,0𝑎³_1,1𝑏_7,3−35𝑎⁷_1,1𝑏_9,1𝑎_2,0+120𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+35𝑎⁸_1,1𝑏_10,0−40𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2+32𝑎⁶_2,0𝑎²_1,1𝑏_8,2+128𝑏_10,0𝑎⁸_2,0−70𝑎²_2,0𝑎⁶_1,1𝑏_10,0+30𝑎²_2,0𝑎⁶_1,1𝑏_8,2−160𝑎⁶_2,0𝑎²_1,1𝑏_10,0−80𝑎⁵_2,0𝑎³_1,1𝑏_9,1−20𝑎³_2,0𝑎⁵_1,1𝑏_7,3+60𝑎³_2,0𝑎⁵_1,1𝑏_9,1+64𝑎_1,1𝑏_9,1𝑎⁷_2,0+8𝑎⁴_2,0𝑎⁴_1,1𝑏_6,4𝑡⁸+240𝑎³_2,0𝑎⁵_1,1𝑏_8,2+128𝑎⁷_2,0𝑎_1,1𝑏_8,2−256𝑎⁵_2,0𝑎³_1,1𝑏_8,2+640𝑎⁵_2,0𝑎³_1,1𝑏_10,0+128𝑎⁸_2,0𝑏_9,1−160𝑎⁴_2,0𝑎⁴_1,1𝑏_7,3−480𝑎⁵_1,1𝑏_10,0𝑎³_2,0+96𝑎⁶_2,0𝑎²_1,1𝑏_7,3+400𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0−512𝑎⁷_2,0𝑏_10,0𝑎_1,1−280𝑎⁶_1,1𝑏_9,1𝑎²_2,0+280𝑎⁷_1,1𝑏_10,0𝑎_2,0+64𝑎⁵_2,0𝑎³_1,1𝑏_6,4−384𝑎⁶_2,0𝑎²_1,1𝑏_9,1𝑡⁷+128𝑎⁸_2,0𝑏_8,2+128𝑏_10,0𝑎⁸_2,0+768𝑎⁶_2,0𝑎²_1,1𝑏_10,0+832𝑎⁵_2,0𝑎³_1,1𝑏_9,1−464𝑎⁵_2,0𝑎³_1,1𝑏_7,3+800𝑎²_2,0𝑎⁶_1,1𝑏_10,0−512𝑎⁶_2,0𝑎²_1,1𝑏_8,2+688𝑎⁴_2,0𝑎⁴_1,1𝑏_8,2−1040𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0−800𝑎³_2,0a⁵_1,1𝑏_9,1+192𝑎⁶_2,0𝑎²_1,1𝑏_6,4+192𝑎⁷_2,0𝑎_1,1𝑏_7,3−384𝑎_1,1𝑏_9,1𝑎⁷_2,0𝑡⁶+−512𝑎⁵_2,0𝑎³_1,1𝑏_10,0+384𝑎⁶_2,0𝑎²_1,1𝑏_9,1−256𝑎⁷_2,0𝑎_1,1𝑏_8,2+128𝑎⁸_2,0𝑏_7,3−544𝑎⁶_2,0𝑎²_1,1𝑏_7,3+768𝑎⁵_2,0𝑎³_1,1𝑏_8,2+256𝑎⁷_2,0𝑎_1,1𝑏_6,4+128𝑎⁸_2,0𝑏_9,1+880𝑎⁵_1,1𝑏_10,0𝑎³_2,0−384𝑎⁷_2,0𝑏_10,0𝑎_1,1−880𝑎⁴_1,1𝑏_9,1𝑎⁴_2,0𝑡⁵+128𝑎⁶_2,0𝑎²_1,1𝑏_8,2+128𝑏_10,0𝑎⁸_2,0−256𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_6,4+128𝑎⁴_2,0𝑎⁴_1,1𝑏_10,0+128𝑎⁸_2,0𝑏_8,2+384𝑎⁶_2,0𝑎²_1,1𝑏_10,0−128𝑎⁷_2,0𝑎_1,1𝑏_7,3−128𝑎⁵_2,0𝑎³_1,1𝑏_9,1𝑡⁴+−128𝑎⁵_2,0𝑎³_1,1𝑏_10,0−256𝑎⁷_2,0𝑏_10,0𝑎_1,1+128𝑎⁶_2,0𝑎²_1,1𝑏_9,1+128𝑎⁸_2,0𝑏_9,1+128𝑎⁸_2,0𝑏_7,3−128𝑎⁷_2,0𝑎_1,1𝑏_8,2𝑡³+128𝑎⁶_2,0𝑎²_1,1𝑏_10,0+128𝑏_10,0𝑎⁸_2,0−128𝑎_1,1𝑏_9,1𝑎⁷_2,0+128𝑎⁸_2,0𝑏_8,2𝑡²+128𝑎⁸_2,0𝑏_9,1−128𝑎⁷_2,0𝑏_10,0𝑎_1,1𝑡+128𝑏_10,0𝑎⁸_2,0. (A.5)

Acknowledgments

The authors are very grateful to the referees for the useful remarks and suggestions. This paper is partially supported by a National Key Basic Research of China (Grant no. 2004CB318000).