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end{aligned} This proves the first formula of Lemma 1.
This proves the first formula of Lemma 2.
Step 4. By the addition formula of Lemma 2.1 we obtain the general meromorphic solutions w ( z − z 0 ).
So, by substituting (sin(frac{pi a}{q})) for x in the second formula of Lemma 2 and making the summation for a with (0leq a leq q-1), with the help of Lemma 1 and Lemma 3, we deduce Theorem 3 immediately.
According to the formulas of Lemma 2.1, the singular values of the origin of system (2.9) are obtained by computing carefully.
According to the formulas of Lemma 3.2, we can find the period constants of four symmetrical singular points and the origin for system (1.2), namely the following several theorems.
end{aligned} This proves formula (II) of Lemma 1.
Now we prove formula (II) of Lemma 1.
end{aligned} (3) From (2) and (3) we may immediately deduce formula (I) of Lemma 1.
Combining the above identity and (16), we find the following generalized formula of [11], Lemma 2.3 (or [8], Eq. (3.4)).
In this paper, we adopt the formula of the solution in Lemma 2.4, which comes from [40].
