Involutivity and Condition W suffice for integrability of split Fréchet tangent distributions, producing a unique maximal foliation, with a dual formulation via annihilating forms.
Pelletier,An integrability criterion for a projective limit of Banach distributions, J
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A Frobenius Theorem on Fr\'echet Manifolds
Involutivity and Condition W suffice for integrability of split Fréchet tangent distributions, producing a unique maximal foliation, with a dual formulation via annihilating forms.