  This note is a continuation of our earlier articles arXiv:1612.08897 and
arXiv:1709.09030, where using the dependent coordinates the local
Lagrange-Poincar\'e equations were obtained for a mechanical system with
symmetry describing the motion of two interacting scalar particles on a special
Riemannian manifold (the product of the total space of the principal fiber
bundle and vector space), on which a free proper and isometric action of a
compact semisimple Lie group is given. Assuming the existence of the parametric
representations for local sections in the principal bundle, we make the
transition to independent coordinates in the obtained Lagrange-Poincar\'e
equations.
