For the first time, we carry out molecular dynamics (MD) simulation of shear-induced martensitic phase transitions between the orthorhombic and non-orthorhombic (triclinic and monoclinic) phases of crystalline polyethylene (PE) in the framework of a realistic all atom model of the polymer. We show that the variation of the shear rate allows observing on a nano-sample both a strongly nonequilibrium phase transition occurring by random nucleation and irregular growth of a new phase ('civilian' way, for rapid deformations) and the coherent, or 'military', kinetics (generally considered as usual for martensitic transformations). We induce transitions from the orthorhombic to the triclinic phase according to two transformation modes observed in experiment on PE single crystals. Rapid deformation favors the transition directly to the triclinic phase, slow deformation -first to the intermediate monoclinic, and only then -to the triclinic phase. The second way corresponds to the experiment on extended chain PE. We explain this result and analyze the competition between different transformation and plastic deformation modes. Rotations of PE chains around their axes necessary for the transition between the orthorhombic and non-orthorhombic phases are executed by short twist defects diffusing along the chains. The transition between the monoclinic and triclinic phases occurs through half-chain-period translations of the chains along their axes, mostly collectively, as crystallographic slips.