The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications

The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus $3$ is described in terms of the gradient of its sigma function. As an application, solutions of the corresponding families of polynomial dynamical systems in $C^4$ with two polynomial integrals are constructed. These systems were introduced by Buchstaber and Mikhailov on the basis of commuting vector fields on the symmetric square of algebraic curves.