  The field of meromorphic functions on a sigma divisor of a hyperelliptic
curve of genus $3$ is described in terms of the gradient of it's sigma
function. Solutions of corresponding families of polynomial dynamical systems
in $\mathbb{C}^4$ with two polynomial integrals are constructed as an
application. These systems were introduced in the work of V. M. Buchstaber and
A. V. Mikhailov on the basis of commuting vector fields on the symmetric square
of algebraic curves.
