Path integral on a manifold with a non-free group action

The method of the factorization of the path integral measure, based on a nonlinear filtering equation, is extended to the case of a nonfree isometric action of the compact semisimple unimodular Lie group on a smooth compact Riemannian manifold. The method is applied to the path integral which describes the "quantum" motion of the scalar particle on this manifold. The relation between path integral representing the solution of the parabolic equation on initial and reduced manifold is derived. It is shown that reduction reduction leads to the non invariance of the path integral measure.