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We describe projective structures on a Riemann surface corresponding tomonodromy groups which have trivial $SL(2)$ monodromies around singularitiesand trivial $PSL(2)$ monodromies along homologically non-trivial loops on aRiemann surface. We propose a natural higher genus analog of Stieltjes-Betheequations. Links with branched projective structures and with Hurwitz spaceswith ramifications of even order are established. We find a higher genus analogof the genus zero Yang-Yang function (the function generating accessoryparameters) and describe its similarity and difference with Bergmantau-function on the Hurwitz spaces.

Stieltjes-Bethe equations in higher genus and branched coverings with even ramifications