Abstract
The purpose of this paper is to relate the non-existence of polynomial integrals for a Hamiltonian system to the breakdown phenomenon of smooth solutions in quasi-linear equations. Using this relation it is shown that for the classical Hamiltonian system with 1.5 degrees of freedom there are no non-trivial third power integrals of motion. The main tool used in the proof is the Lax analysis on formation of singularities in quasi-linear equations. Some results and perspectives for the case of higher degrees are discussed.