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In the approach to the geometric quantization, based on the conversion ofsecond-class constraints, we resolve the respective non-linear zero curvatureconditions for the extended symplectic potential. From the zero curvatureconditions, we deduce new, linear, equations for the extended symplecticpotential. Then we show that being the linear equations satisfied, theirsolution does certainly satisfy the non-linear zero curvature condition, aswell. Finally, we give the functional resolution to the new linear equations,and then deduce the respective path integral representation. We do ourconsideration as to the general case of a phase superspace where both Boson andFermion coordinates are present on equal footing.

Conversion of second-class constraints and resolving the zero curvature conditions in the geometric quantization theory