Sergey Tsarev (Siberian Federal University, Krasnoyarsk, Russia)
  Geodesic flows with polynomial and non-polynomial integrals.

A new algorithmic method for local construction of two-dimensional metrics with polynomial or non-polynomial (rational or more general) first integral is presented. The method allows to produce multiparametric families of such metrics and its integrals. We conjecture that locally two-dimensional metrics posessing first integrals are dense in the functional space of all local two-dimensional metrics.