Matrix-second order differential equations and chaotic Hamiltonian systems

We consider matrix-second order differential equations which are perturbations of the harmonic flow on the space of matrices. Experimental evidence of the non integrability of the two degrees of freedom Hamiltonian system provides an indication of the non existence of a Lax pair with commuting eigenvalues for perturbations of order six. This shows the specificity of quartic perturbations for which such a Lax pair was precedently obtained.Supported by Istituto Nazionale di Alta Matematica F. Severi. This article has been written while the second author was visiting ETH.