On the two-gap locus for the elliptic Calogero-Moser model

We give an analytical description of the locus of the two-gap elliptic potentials associated with the corresponding flow of the Calogero-Moser system. We start with the description of Treibich-Verdier two-gap elliptic potentials. The explicit formulae for the covers, wavefunctions and Lame polynomials are derived, together with a new Lax representation for the particle dynamics on the locus. We then consider more general potentials within the Weierstrass reduction theory of theta functions to lower genera. The reduction conditions in the moduli space of the genus-2 algebraic curves are given. This is a subvariety of the Humbert surface, which can be singled out by the condition of the vanishing of some theta constants.