ABOUT GEOMETRIZATION OF THE DYNAMICS*) D. BALEANU**) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Moscow region, Russia and Middle East Technical University, Physics Department, 06531 Ankara, Turkey A.(KALKANLI) KARASU Middle East Technical University, Physics Department, 06531 Aniara, Turkey N. MAKHALDIANI Laboratory of Computing Techniques and Automation, Joint Institute for Nuclear Research, Dubna, Moscow region, Russia Received 3 August 1999 The connection between Killing tensors and Lax operators are presented. The Toda lattice case and the Rindler system are analyzed in details. *) Presented at the 8th Colloquium "Quantum groups and integrable systems", Prague, 17-19 June 1999. **) Permanent address: Institute of Space Sciences, P.O.Box MG-23, R 76900 Magurele- Bucharest, Romania Czechoslovak Journal of Physics, Vol. 50 (2000), No. 1 17 In [4] it was shown that for a theory with a dual Poisson bracket structure, the sufficient condition for integrability is the vanishing of the Nijenhuis tensor. Another method to construct integrals of motion, for the finite systems, is to use the Jacobi geometry and to analyze the existence of Killing tensor in the case of geodesic motion. The geodesic equations are We can construct the conserved quantities 1 Introduction It is well known that, many of the classical dynamical theories are completely integrable [1-2]. These are both finite as well as infinite dimensional theories which describe physical systems of importance. For discrete finite systems, it is known that the zero Nijenhuis tensor [3] condition can be used to construct conserved quantities in involution [4], Let us note that for a given (1,1) tensor such as Svu we can define the Nijenhuis torsion tensor as [3]