This paper contributes to the study of Engel structures and their classification. The main result introduces the notion of a loose family of Engel structures and shows that two such families are Engel homotopic if and only if they are formally homotopic. This implies a complete $h$-principle when auxiliary data is fixed. As a corollary, we show that Lorentz and orientable Cartan prolongations are classified up to homotopy by their formal data.

Let Ω = (ω1,…,ωn-k) be differential 1-forms with polynomial coefficients in Rn. A Pfaffian manifold of Ω is by definition a maximal integral k-manifold of Ω. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of Ω is finite and, moreover, bounded by a computable function in variables n, k and the degree of ω1,…, ωn-k. Finiteness is proved also in any o-minimal structure.

We give also an example of a semi-algebraic C1 differential form on a semi-algebraic C2 3-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).

We discuss the Monge problem for under-determined systems of ordinary differential equations with an arbitrary degree of freedom and give a sufficient condition, in terms of truncated multi-flag systems, for the Monge property to hold. This condition extends in a natural way the Cartan criterion valid for systems with one degree of freedom.