Equivalent approaches to the concept of completeness of foliations with transverse linear connection

We prove the equivalence of three different approaches to the definition of completeness of a foliation with transverse linear connection. It is shown that for the transverse affine foliations $(M, F)$ of codimension $ q, \, q \geqslant 1,$ each of the mentioned above conditions are equivalent to fulfillment of the following two conditions: 1) there exists an Ehresmann connection to $ (M, F)$; 2) the induced foliation on the universal covering space is formed by fibres of submersion onto $q$-dimensional affine space.