  The aim of this paper is to study from the point of view of linear
connections the data $(M,\mathcal{D},g,W),$ with $M$ a smooth $(n+p)$
dimensional real manifold, $(\mathcal{D},g)$ a
\textit{$n$}\textit{\emph{dimensional semi-Riemannian distribution}}\emph{}on
$M,$ $\mathcal{G}$ the conformal structure generated by $g$ and $W$ a Weyl
substructure: a map $W:$ $\mathcal{G}\to$ $\Omega^{1}(M)$ such that
$W(\overline{g})=W(g)-du,$ $\overline{g}=e^{u}g;u\in C^{\infty}(M)$. Compatible
linear connections are introduced as a natural extension of similar notions
from Riemannian geometry and such a connection is unique if a symmetry
condition is imposed. In the foliated case the local expression of this unique
connection is obtained. The notion of Vranceanu connection is introduced for a
pair (Weyl structure, distribution) and it is computed for the tangent bundle
of Finsler spaces, particularly Riemannian, choosing as distribution the
vertical bundle of tangent bundle projection and as 1-form the Cartan form.
