  The question of finding solutions to given implicit differential equations
(IDE) has been answered by several authors in the last few years, using
different approaches, in an algebraic and also a geometric setting. Many of
those results assume in one way or another that the subimmersion theorem can be
applied at several stages of the reduction algorithm, which, roughly speaking,
allows to reduce a given IDE to a collection of ODE depending on parameters.
The main purpose of the present paper is to improve some of the known results
by introducing at each stage of the reduction algorithm a desingularization of
the manifolds with singularities that may appear when the subimmersion theorem
cannot be applied. This can be done for analytic IDE by using some fundamental
results on subanalytic subsets and desingularization of closed subanalytic
subsets due mainly to Lojasiewicz, Hironaka, Gabrielov, Hardt, Bierstone,
Milman and Sussmann, among others. We will show how this approach helps to
understand the dynamics given by the Lagrange-D'Alembert-Poincare equations for
the symmetric elastic sphere.
