Parametric Korteweg–de Vries hierarchy and hyperelliptic sigma functions

In this paper, a parametric Korteweg–de Vries hierarchy is defined that depends on an infinite set of graded parameters $a = (a_4,a_6,\dots)$. It is shown that, for any genus $g$, the Klein hyperelliptic function $\wp_{1,1}(t,\lambda)$ defined on the basis of the multidimensional sigma function $\sigma(t, \lambda)$, where $t = (t_1, t_3,\dots, t_{2g-1})$ and $\lambda = (\lambda_4, \lambda_6,\dots, \lambda_{4 g + 2})$, specifies a solution to this hierarchy in which the parameters $a$ are given as polynomials in the parameters $\lambda$ of the sigma function. The proof uses results concerning the family of operators introduced by V. M. Buchstaber and S. Yu. Shorina. This family consists of $g$ third-order differential operators in $g$ variables. Such families are defined for all $g \geqslant 1$, the operators in each of them pairwise commute with each other and also commute with the Schrödinger operator. In this paper a relationship between these families and the Korteweg–de Vries parametric hierarchy is described. A similar infinite family of third-order operators on an infinite set of variables is constructed. The results obtained are extended to the case of such a family.