  For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a
transversal distribution $\Delta$ is defined by the torsion tensor of the web.
In general, this distribution is not integrable. The authors find necessary and
sufficient conditions of its integrability and prove the existence theorem for
webs W (3, 2, 2) with integrable distributions $\Delta$.
  They prove that for a web W (3, 2, 2) with the integrable distribution
$\Delta$, the integral surfaces $V^2$ of $\Delta$ are totally geodesic in an
affine connection of a certain bundle of affine connections.
  They also consider a class of webs W (3, 2, 2) for which the integral
surfaces $V^2$ of $\Delta$ are geodesicly parallel with respect to the same
affine connections and a class of webs for which two-dimensional webs W (3, 2,
1) cut by the foliations of W (3, 2, 2) on $V^2$ are hexagonal. They prove the
existence theorems for webs of the latter class as well as for webs of the
subclass which is the intersection of two classes mentioned above. The authors
also establish relations between three-webs considered in the paper.
