The Sylvester equation and the elliptic Korteweg-de Vries system

The elliptic Korteweg-de Vries (KdV) system is a multi-component generalization of the lattice potential KdV equation, whose soliton solutions are associated with an elliptic Cauchy kernel (i.e., a Cauchy kernel on the torus). In this paper we generalize the class of solutions by using a Sylvester type matrix equation and rederiving the system from the associated Cauchy matrix. Our starting point is the Sylvester equation in the form of $~\boldsymbol{k} \boldsymbol{M}+ \boldsymbol{M} \boldsymbol{k} = \boldsymbol{r} {\boldsymbol{c}}^{T}-g\boldsymbol{K}^{-1} \boldsymbol{r} {\boldsymbol{c}}^{T} \boldsymbol{K}^{-1}$ where $\boldsymbol{k}$ and $\boldsymbol{K}$ are commutative matrices and obey the matrix relation ${\boldsymbol{k}}^2=\boldsymbol{K}+3e_1\boldsymbol{I}+g{\boldsymbol{K}}^{-1}$. The obtained elliptic equations, both discrete and continuous, are formulated by the scalar function $S^{(i,j)}$ which is defined using $(\boldsymbol{k},\boldsymbol{K}, \boldsymbol{M}, \boldsymbol{r},\boldsymbol{c})$ and constitute an infinite size symmetric matrix. Lax pairs for both the discrete and continuous system are derived. The explicit solution $\boldsymbol{M}$ of the Sylvester equation and generalized solutions of the obtained elliptic equations are presented according to the canonical forms of matrix $\boldsymbol{k}$.