On Hamiltonian Dynamical Systems with Kinetic Energy Depending on Momenta Modules

We consider a new class of Hamiltonian dynamical systems with two degrees of freedom whose kinetic energy is a function of momenta's modules. Equations of motion for such systems are easily integrated in each of succesive time intervals; thus, in principle, all the trajectories can be found explicitly. A Poincare mapping for such systems can be reduced to a mapping of the least positive root for a system of transcendental (in the general case) equations.
On the other hand, dynamical systems of this type exhibit a number of properties typical to non-intergable systems (e.g., an existence of stable and unstable periodic orbits, their bifurcations, creation of stochastic layers in vicinity of destroyed separatrices, regions of global chaotic motion, etc.).
As an example, a system with a simple potential that is quadratic in both coordinates is studied.