  We consider the procedure that constructs log Calabi-Yau compactifications of
weak Landau-Ginzburg models of Fano varieties. We apply it for del Pezzo
surfaces and coverings of projective spaces of index one. For the coverings of
degree greater then 2 the log Calabi-Yau compactification is singular;
moreover, no smooth projective log Calabi-Yau compactification exists. We also
prove in the cases under consideration the conjecture saying that the number of
components of the fiber over infinity %and of finite fibers is equal to the
dimension of an anticanonical system of the Fano variety.
