  In this paper we exploit the use of symmetries of a physical system so as to
characterize algebraically the corresponding solution manifold by means of
Noether invariants. This constitutes a necessary preliminary step towards the
correct quantization in non-linear cases, where the success of Canonical
Quantization is not guaranteed in general. To achieve this task "point
symmetries" of the Lagrangian are generally not enough, and the notion of
contact transformations is in order: the solution manifold can not be in
general parametrized by means of Noether invariants associated with basic point
symmetries. The use of the contact structure given by the Poincar\'e-Cartan
form permits the definition of the symplectic form on the solution manifold,
through some sort of Hamilton-Jacobi transformation. It also provides the
required basic symmetries, realized as Hamiltonian vector fields associated
with global functions on the solution manifold (thus constituting an inverse of
the Noether Theorem), lifted back to the evolution space through the inverse of
this Hamilton-Jacobi mapping. In this framework, solutions and symmetries, as a
whole, are somehow identified and this correspondence is also kept at a
perturbative level. We present non-trivial examples of this interplay between
symmetries and solutions pointing out the usefulness of this mechanism in
approaching the corresponding quantization. In particular, we achieve the
proper generalization of the Heisenberg-Weyl algebra for the non-linear
particle sigma model in $S^3$ within this framework, and notice that a subset
of the classical symmetries corresponding with this quantizing algebra (those
generalizing boosts) are necessarily of non-point character.
