Finite-gap solutions of Abelian Toda chain of genus 4 and 5 in elliptic functions

A reduction theorem is formulated and proved. Smooth real solutions of the abelian Toda lattice of the genus 4 and 5 are obtained in terms of the elliptic functions. In terms of the $g$-dimensional theta-functions the solutions of the genus $2g$ and $2g+1$ are constructed for the discrete Peierls–Fröhlich model in the absence of intramolecular deformation.