Bol transformation quasigroups and three-webs defined by them

The notions of a local smooth quasigroup and a quasigroup of transformations are natural generalizations of the notions of the Lie group and the Lie transformation group. We define the quasigroup of transformations as an action $f$ of the local smooth $q$-dimensional quasigroup $Q(*)$ on the smooth $p$-dimensional manifold $Y$ $(1\leq p\leq q)$ given by a smooth function
$$ f\colon Q\times Y\to Y,\quad z=f(a,y),\quad a\in Q,\quad y,z\in Y. $$
On the other hand, the equation $z=f(a,y)$ defines the three-web $QW(p,q,q)$ formed by a foliation of $p$-dimensional leaves $a=\mathrm{const}$ and two foliations of $q$-dimensional leaves $y=\mathrm{const}$ and $z=f(a,y)=\mathrm{const}$ on the manifold $Q\times Y$. Thus, we can use the three-web theory methods to study different classes of smooth local quasigroups of transformations. In the present paper, we investigate Bol quasigroups of transformations characterized by some condition on the function $f$.