Given a system of smooth 1-forms $ heta$ = ($ heta^1$,...,$ heta^s$) on a smooth manifold $M^m$, we give a necessary and sufficient condition for M to be foliated by integral manifolds of dimension n, n $leq$ p := m - s, and construct an integrable supersystem ($ heta,eta$) by finding additional 1-forms $eta$ = ($eta^1$,...,$eta^{p-n}$). We also give a necessary and sufficient condition for M to be foliated by reduced submanifolds of dimension n, n $geq$ p, and construct an integrable subsystem ($d ho^1$,...,$d ho^{m-n}$) by finding a system of first integrals $ ho=( ho^1$,...,$ ho^{m-n})$. The special case n = p is the Frobenius theorem on involutivity.