On first integrals of Hamiltonian systems on the 2-torus. 

   We consider Hamiltonian systems related to geodesic flows (including magnetic ones) and natural mechanical systems on the 2-torus and study the question of their integrability. Generally speaking, this problem can be reduced to the search for an additional first integral which is independent on the Hamiltonian. The different questions related to local and global existence of such integrals, polynomial or rational in momenta, will be discussed. 
   This talk is based on joint results with M. Bialy, A.E. Mironov, A.A. Valyuzhenich.