Given a system of s independent 1-forms on a smooth manifold M of dimension m, we study the existence of integral manifolds by means of various generalized versions of the Frobenius theorem. In particular, we present necessary and sufficient conditions for there to exist s -parameter (s < s) family of integral manifolds of dimension p := m−s, and a necessary and sufficient condition for there to exist integral manifolds of dimension p , p ≤ p. We also present examples and applications to complex analysis in several variables.