  In this paper we investigate the existence of ``partially'' isometric
immersions. These are maps f:M->R^q which, for a given Riemannian manifold M,
are isometries on some sub-bundle H of TM. The concept of free maps, which is
essential in the Nash--Gromov theory of isometric immersions, is replaced here
by that of H-free maps, i.e. maps whose restriction to H is free. We prove,
under suitable conditions on the dimension q of the Euclidean space, that
H-free maps are generic and we provide, for the smallest possible value of q,
explicit expressions for H-free maps in the following three settings:
1-dimensional distributions in R^2, Lagrangian distributions of completely
integrable systems, Hamiltonian distributions of a particular kind of Poisson
Bracket.
