  Using the path integral measure factorization method based on the nonlinear
filtering equation from the stochastic process theory, we consider the
reduction procedure in Wiener path integrals for a mechanical system with
symmetry that describes the motion of two interacting scalar particles on a
special smooth compact Riemannian manifold - the product of total space of the
principal fiber bundle and the vector space. The original manifold, the
configuration space of this system, is endowed with an isometric free proper
action of a compact semisimple unimodular Lie group. The proposed reduction
procedure leads to the integral relation between path integrals that represent
fundamental solutions of the inverse Kolmogorov equations on the initial and
reduced manifolds. For the case of reduction onto the zero-momentum level, the
reduction Jacobian is obtained, which is an additional potential term to the
Hamiltonian.
