2-dimensional finite-gap Schr\"{o}dinger operator whose spectrum admits two involutions

Two-dimensional Schr\"{o}dinger operators that are finite-gap at one energy level are introduced in 1976 by Dubrovin, Krichever and Novikov. In two subsequent works by Novikov and Veselov the potentiality conditions for them have been studied, that are conditions for the magnetic term to be absend. Besides their physical importance, these works played a crucial role in solving out the Riemann--Shottki problem of indetification of Prymians of smooth coverings with two branch points in the class of principally polarized Abelian varieties, going back to 80s, and completed in Krichever'06, and Krichever and Grushevsky'07. For smooth coverings with more than two branch points, the Prym varieties are no longer principally polarized. In wellknown Fay's lectures, certain isogenic to them principally polarized varieties are introduced, which we refer to as isoPrymians. In the present work we propose the new potentiality conditions for the Schr\"{o}dinger operators in question, and a related approach to identification of isoPrymians of smooth double coverings of curves of a certain class, with more than two branch points.