From Theorem 3.2 in [3], we have

**Lemma 3.1** *For system* {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0}, *we can determine uniquely an extended formal power series*

F(z,w)=zw\sum _{k=0}^{\mathrm{\infty}}\frac{{f}_{5k}(z,w)}{{(zw)}^{2k}},

(3.1)

*such that*

\frac{dF}{dT}{|}_{{\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0}}=\sum _{m=1}^{\mathrm{\infty}}{\mu}_{m}^{(0)}{(zw)}^{m+2},

(3.2)

*where* {c}_{00}=1, {c}_{kk}=0, k=1,2,\dots , {f}_{5k}(z,w)={\sum}_{\alpha +\beta =5k}{c}_{\alpha \beta}{z}^{\alpha}{w}^{\beta}, *and for any positive integer* *m*, *the* *mth singular point quantity at the origin* {\mu}_{m}^{(0)} *can be determined by the following recursion formulas*:

Using the recursion formulas to do symbolic computation, we have

**Theorem 3.1** *The first nine singular point quantities at the origin of system* {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0} *are as follows*:

*In the above expression of* {\mu}_{k}^{(0)}, *we have already let* {\mu}_{1}^{(0)}=\cdots ={\mu}_{k-1}^{(0)}=0, k=2,3,\dots ,9.

From Theorem 3.1, it is easy to get

**Theorem 3.2** *For system* {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0}, *the first nine singular point quantities at the origin are zero if and only if one of the following conditions holds*:

\begin{array}{rl}(\mathrm{I})& {a}_{21}={b}_{21},\phantom{\rule{2em}{0ex}}{b}_{30}=\frac{1}{3}{a}_{12},\phantom{\rule{2em}{0ex}}{a}_{30}=\frac{1}{3}{b}_{12},\phantom{\rule{2em}{0ex}}{a}_{12}{b}_{12}\ne 0;\\ (\mathrm{II})& {a}_{21}={b}_{21},\phantom{\rule{2em}{0ex}}{a}_{12}{a}_{30}={b}_{12}{b}_{30},\phantom{\rule{2em}{0ex}}{a}_{12}^{2}{b}_{03}={b}_{12}^{2}{a}_{03},\\ {a}_{12}{b}_{30}{b}_{03}={b}_{12}{a}_{30}{a}_{03},\phantom{\rule{2em}{0ex}}{b}_{30}^{2}{b}_{03}={a}_{30}^{2}{a}_{03}.\end{array}

*Proof* Putting condition (I) or (II) into expression (3.5) can easily complete the proof of sufficiency.

Now, let us prove the necessity. If {a}_{12}={b}_{12}=0, then {\mu}_{2}^{(0)}={\mu}_{4}^{(0)}={\mu}_{5}^{(0)}={\mu}_{6}^{(0)}={\mu}_{7}^{(0)}={\mu}_{8}^{(0)}={\mu}_{9}^{(0)}=0, {\mu}_{3}^{(0)}=-\frac{9({a}_{03}{a}_{30}^{2}-{b}_{03}{b}_{30}^{2})}{8{\lambda}^{2}}. Therefore, {\mu}_{k}^{(0)}=0 (k=1,2,\dots ,9) yields

{a}_{21}={b}_{21},\phantom{\rule{2em}{0ex}}{a}_{30}^{2}{a}_{03}={b}_{30}^{2}{b}_{03},\phantom{\rule{2em}{0ex}}{a}_{12}={b}_{12}=0.

(3.6)

If {a}_{12}{b}_{12}\ne 0, notice that {\mu}_{1}^{(0)}=-{a}_{21}+{b}_{21}=0, {\mu}_{2}^{(0)}=\frac{{a}_{12}{a}_{30}-{b}_{12}{b}_{30}}{\lambda}=0, there exist {r}_{32} and *p* such that

{a}_{21}={b}_{21}={r}_{21},\phantom{\rule{2em}{0ex}}{b}_{30}=p{a}_{12},\phantom{\rule{2em}{0ex}}{a}_{30}=p{b}_{12}.

(3.7)

Substituting (3.7) into expression (3.5), we get

\begin{array}{r}{\mu}_{3}^{(0)}=\frac{({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})(-1+3p)(1+3p)}{8{\lambda}^{2}},\\ {\mu}_{4}^{(0)}=\frac{({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})(-1+3p){r}_{21}}{6{\lambda}^{3}},\\ {\mu}_{5}^{(0)}=\frac{({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})(-1+3p)(-27{a}_{03}{b}_{03}+16{a}_{12}{b}_{12}+216\lambda )}{648{\lambda}^{4}},\\ {\mu}_{7}^{(0)}=-\frac{({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})(405{a}_{03}^{2}{b}_{03}^{2}-8\text{,}928{a}_{03}{a}_{12}{b}_{03}{b}_{12}+6\text{,}400{a}_{12}^{2}{b}_{12}^{2})(-1+3p)}{466\text{,}560{\lambda}^{6}},\\ {\mu}_{8}^{(0)}=-\frac{7(-45{a}_{03}{b}_{03}+32{a}_{12}{b}_{12})({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})({a}_{12}^{2}{b}_{03}+{a}_{03}{b}_{12}^{2})(-1+3p)}{69\text{,}984{\lambda}^{7}},\\ {\mu}_{9}^{(0)}=-\frac{11{a}_{12}^{2}{b}_{12}^{2}(-224\text{,}181{a}_{03}{b}_{03}+164\text{,}000{a}_{12}{b}_{12})({a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2})(-1+3p)}{13\text{,}778\text{,}100{\lambda}^{8}}.\end{array}

(3.8)

405{a}_{03}^{2}{b}_{03}^{2}-8\text{,}928{a}_{03}{a}_{12}{b}_{03}{b}_{12}+6\text{,}400{a}_{12}^{2}{b}_{12}^{2}=0 and -224\text{,}181{a}_{03}{b}_{03}+164\text{,}000{a}_{12}{b}_{12}=0 do not simultaneously hold, thus from {\mu}_{2}^{(0)}={\mu}_{7}^{(0)}={\mu}_{9}^{(0)}=0, we have (i) p=\frac{1}{3} or (ii) {a}_{12}{a}_{30}-{b}_{12}{b}_{30}=0, {a}_{12}^{2}{b}_{03}-{a}_{03}{b}_{12}^{2}=0. At this moment, {\mu}_{3}^{(0)}={\mu}_{4}^{(0)}={\mu}_{5}^{(0)}={\mu}_{8}^{(0)}=0. Condition (i) or (ii) combining with {\mu}_{1}^{(0)}=0 implies the following conditions, respectively:

Condition (3.6) plus (3.10) is equivalent to condition (II). The proof is completed. □

From the definition of elementary Lie invariants in [17], we can obtain

**Lemma 3.2** *All the elementary Lie invariants of system* {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0} *are as follows*:

\begin{array}{r}\lambda ,\phantom{\rule{2em}{0ex}}{a}_{21},\phantom{\rule{2em}{0ex}}{b}_{21},\phantom{\rule{2em}{0ex}}{a}_{30}{b}_{30},\phantom{\rule{2em}{0ex}}{a}_{12}{b}_{12},\\ {a}_{03}{b}_{03},\phantom{\rule{2em}{0ex}}{a}_{12}{a}_{30},\phantom{\rule{2em}{0ex}}{b}_{12}{b}_{30},\\ {a}_{12}^{2}{b}_{03},\phantom{\rule{2em}{0ex}}{a}_{12}{b}_{30}{b}_{03},\phantom{\rule{2em}{0ex}}{b}_{30}^{2}{b}_{03},\\ {b}_{12}^{2}{a}_{03},\phantom{\rule{2em}{0ex}}{b}_{12}{a}_{30}{a}_{03},\phantom{\rule{2em}{0ex}}{a}_{30}^{2}{a}_{03}.\end{array}

(3.11)

**Theorem 3.3** *For system* {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0}, *all the singular point quantities at the origin are zero if and only if the first nine singular point quantities are zero*, *i*.*e*., *one of the two conditions in Theorem * 3.2 *holds*. *Correspondingly*, *the two conditions in Theorem * 3.2 *are the center conditions of the origin*.

*Proof* When condition (I) holds, system {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0} has the integrating factor M(z,w)={(zw)}^{-5}; when condition (II) holds, system {\text{(2.10)}}_{{\delta}_{1}={\delta}_{2}=0} satisfies the conditions of the extended symmetric principle (Theorem 2.6 in [17]). □