  We consider a 3-dimensional smooth manifold $M$ equipped with an arbitrary,
\textit{a priori} non-integrable, distribution (plane field) ${\cal D}$ and a
vector field $T$ transverse to ${\cal D}$. Using a 1-form $\omega$ such that
${\cal D} = \ker\,\omega$ and $\omega(T)=1$ we construct a 3-form analogous to
that defining the Godbillon-Vey class of a foliation, and show how does this
form depend on $\omega$ and~$T$. For a compatible Riemannian metric on $M$, we
express this 3-form in terms of the curvature and torsion of normal curves and
the non-symmetric second fundamental form of ${\cal D}$. We deduce
Euler-Lagrange equations of associated functionals: for variable $({\cal D},T)$
on $M$, and for variable Riemannian or Randers metric on $(M,{\cal D})$. We
show that for a geodesic field $T$ (e.g., for a contact structure) such $({\cal
D},T)$ is critical, characterize critical pairs when ${\cal D}$ is integrable,
and prove that these critical pairs are not extrema.
