Methods of weyl representation of the phase space and canonical transformations. II

Nonlinear canonical transformations realized in the space of Weyl symbols of quantum operators are studied. The kernels of the transformations, the symbol of the intertwining operator of the group of inhomogeneous point transformations, and the group characters are constructed. The group of $\mathbf{PL}$ transformations, which is the free product of the group of point, $\mathbf{P}$ and linear, $\mathbf{P}$ transformations is considered. The simplest $\mathbf{PL}$ complexes relating problems with different potentials, in particular, containing a general Darboux transformation of the factorization method, are constructed. The kernel of an arbitrary element of the group $\mathbf{PL}$ is found.