The theory of the pairs of structures is well applied in the case of the generalised Riemann spaces and Romanian geometrist had remarkable achievements in this field. In order to solve this problem, professor R. Miron and Gh. Atanasiu have imposed the essential condition of permutability of Obata operators attached to the metrical and almost simplistic structures of the Riemann - Einsenhart space. In this paper we will take into consideration the problem of geometric structure pairs, defined on the total space of a vector bundle from another point of view, i.e. a pair of structures is given by both a horizontal and vertical structure, these structures being called $(h,v)$-structures. No such structures were studied so far. The determination of the compatible connections with such structures on $E$ is simpler then the determination of the connections simultaneously compatible with two defined structures on $E$.