us to present effective solutions of this problem for Abelian functions of the hyperelliptic curve. It is noteworthy that because integrals of motion can be expressed in terms of second kind periods in this place, the results of Section 2 are required. The results obtained in Section 3 are exemplified in details by curves of genera one and two. All consideration is based on explicit uniformization of the space of universal bundles of the hyperelliptic Jacobian.

Modular representation of periods of hyperelliptic co-homologies
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Modular invariance of the Weierstrass elliptic $\sigma$-function, $\sigma=\sigma(u;g_2,g_3)$ follows from its defining in in [@wei885] in terms of recursive series in terms of variables $(u,g_2,g_3)$. Alternatively $\sigma$-function can be represented in terms of Jacobi $\theta$-function and its modular invariance follows from transformation properties of $\theta$-functions. The last representation involves complete elliptic integrals of first and second kind and their representations in terms of $\theta$-constants are classically known. In this section we are studying generalizations of these representations to hyperelliptic curves of higher genera realized in the form $$\begin{aligned}
 y^2 = P_{2g+1}(x)=   (x-e_1)
\cdots (x-e_{2g+1}) \label{HCurve} \end{aligned}$$ Here $P_{2g+1}(x)$ - monic polynomial of degree $2g+1$, $e_i\in \mathbb{C}$ - branch points and the curve supposed to be non-degenerate, 