Schrödinger equations on elliptic curves: symmetries, solutions and eigenvalue problem

Abstract: In this paper, we study Schr\"{o}dinger equations on elliptic curves called generalized Lam\'{e} equations. We suggest a method of finding integrable potentials for Schr\"{o}dinger type equations. We apply this method to the Lam\'{e} equations and provide a sequence of integrable potentials for which the eigenvalue problem is solved explicitly.