  We prove two new results connected with elliptic Fokker-Planck-Kolmogorov
equations with drifts integrable with respect to solutions. The first result
answers negatively a long-standing question and shows that a density of a
probability measure satisfying the Fokker-Planck-Kolmogorov equation with a
drift integrable with respect to this density can fail to belong to the Sobolev
class~$W^{1,1}(\mathbb{R}^d)$. There is also a version of this result for
densities with respect to Gaussian measures. The second new result gives some
positive information about properties of such solutions: the solution density
is proved to belong to certain fractional Sobolev classes.
