  We introduce a constructive method that provides the local solution of
general implicit systems in arbitrary dimension via Hamiltonian type equations.
A variant of this approach constructs parametrizations of the manifold,
extending the usual implicit functions solution. We also discuss the critical
case of the implicit functions theorem, define the notion of generalized
solution and prove existence and properties. Examples are also indicated. The
applications concern necessary conditions and algorithms in nonconvex
optimization problems and their perturbations.
