The Henon-Heiles system in the general form has been considered. In a nonintegrable case with the help of the Painleve test new solutions have been found as formal Laurent or Puiseux series, depending on three parameters. One of parameters determines a location of the singularity point, other parameters determine coefficients of series. It has been proved, that if absolute values of these two parameters are less or equal to unit, then obtained series converge in some ring. For some values of these parameters the obtained Laurent series coincide with the Laurent series of the known exact solutions.