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We determineÌý topology of the moduli space for Lame functions of degree m.ÌýThis is a Riemann surface which consists of two connectedÌýcomponents when m ≥ 2; we find the Euler characteristics and generaÌýof these components. As a corollary we prove a conjecture of Robert MaierÌýon degrees of Cohn’s polynomials. These results are obtained with theÌýhelp of a geometric description of these Riemann surfaces, as quotientsÌýof the moduli spaces for certain singular flat triangles.ÌýAn application is given to the study of metrics of constant positiveÌýcurvature with one conic singularity with the angle 2Ï€(2m + 1) on aÌýtorus. We show that the degeneration locus of such metrics is a unionÌýof smooth analytic curves andÌý enumerate these curves.