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.