Moduli spaces of spherical tori with one conic singularity

We discuss metrics of constant positive curvature with finitely many conic singularities on compact surfaces. The problem is to describe the space of such metrics with prescribed angles at the singularities.
After a brief survey of the known general results on the problem, we concentrate on the case of tori with one singularity, and describe the space of all such metrics with fixed angle at the singularity. A connection of the problem with Lamé equations will be also explained.
Based on the joint work with Andrei Gabrielov (Purdue University), Gabriele Mondello (Universita di Roma) and Dmitry Panov (Kings College, London).