The structure of foliations with integrable Ehresmann connection

We study foliations of arbitrary codimension $q$ on $n$-dimensional smooth manifolds admitting an integrable Ehresmann connection. The category of such foliations is considered, where isomorphisms preserve both foliations and their Ehresman connections. We show that this category can be considered as that of bifoliations covered by products. We introduce the notion of a canonical bifoliation and we prove that each foliation $(M, F)$ with integrable Ehresmann connection is isomorphic to some canonical bifoliation. A category of triples is constructed and we prove that it is equivalent to the category of foliations with integrable Ehresmann connection. In this way, the classification of foliations with integrable Ehresman connection is reduced to the classification of associated diagonal actions of discrete groups of diffeomorphisms of the product of manifolds. The classes of foliations with integrable Ehresmann connection are indicated. The application to $G$-foliations is considered.