very recently, using the fact that sigma is an entire function in the parameters of the curve (in contrast with theta-function), families of degenerate solutions for solitonic equations were obtained [@bl18; @ben18]. In recent papers [@bm17] and [@bm18], an algebraic construction of a wide class of polynomial Hamiltonian integrable systems was given, and those of them whose solutions are given by hyperelliptic $\wp$-functions were indicated.

Revival of interest in multi-variable $\sigma$-functions is in many respects guided by H.Baker’s exposition of the theory of Abelian functions, which go back to K.Weierstrass and F.Klein and is well documented and developed in his remarkable monographs [@bak998; @bak903]. The heart of his exposition is the representation of fundamental bi-differential of the hyperelliptic curve in algebraic form, in contrast with the representation as a double differential of Riemann theta-function developed by Fay in the monograph, [@fay973]. Recent investigations have demonstrated that Baker’s approach can be extended beyond hyperelliptic curves to wider classes of algebraic curves.

The multi-variable $\sigma$-function of algebraic curve $\mathcal{C}$ is known to be represented in terms of $\theta$-function of the curve as a function of two groups of variables - the Jacobian of the curve, $\mathrm{Jac}(\mathcal{C})$ and the Riemann matrix $\tau$. In 