  Some results of numerical study of the canonical map with a sawtooth force
are given and discovered new unexpected dynamical effects are described. In
particular, it is shown that if the values of the system parameter K belong to
the countable set determined by Ovsyannikov's theorem, separatrices of primary
resonances are not splitted and chaotic layers are not formed. One more set of
values of the parameter related to the other family of nondestructed
separatrices of primary resonances was found. The mechanism explaining the
stability of the primary resonance separatrix in the critical regime is found
and described. First secondary resonances were studies and for them were found
the K-values at which their separatrices are not splitted also. New problems
and open questions occurred in this connections whose solution can facilitate
the further development of the nonlinear Hamiltonian systems theory are
presented.
