On the Integrability of Distributions in $\mathbb{Z}$-graded Geometry

Integrability of distributions on $\mathbb{Z}$-graded manifolds is examined. First, the Local Frobenius theorem – the local existence of flat coordinates for an involutive distribution – is proved. Then, two different notions of an integral submanifold are discussed. It is found that, for the stronger of the two notions, the Global Frobenius theorem holds, but as one implication only: involutivity implies integrability. It is then shown that integrability implies a certain weaker version of involutivity. Simple counterexamples of the converse implications are given.