ABSTRACT
This chapter provides the phenomenon that "there are exactly solvable systems that are not completely integrable". It explores Hamiltonian system completely integrable if there is a holomorphic transformation that reduces the system to action-angle variables or, in the case of compact phase space, the phase space of the system is decomposed into tori of different dimensions. There is a large class of systems of nonlinear differential equations of physical interest that exhibit different kinds of ergodic behavior: transition from "completely integrable" to ergodic and ergodic with strange attractors of different Hausdorff dimensions. A flow topologically defined as a "billiard system" in the fundamental domain cannot possibly be described through a solution of an algebraic system of nonlinear ordinary differential equations. Hence nonlinear differential equations with the same topological properties as "billiard" flows on F appear only after proper transformations analogous to the inverse Abel map from a curve to its Jacobian.
