The generalized Hénon–Heiles system with an additional non-polynomial term is considered. The standard method for the search of the elliptic solutions is a transformation of an initial nonlinear polynomial differential equation into a nonlinear algebraic system. It has been demonstrated that the use of the Laurent-series solutions allows to simplify the resulting algebraic system. This procedure has been automatized and generalized on some type of multivalued solutions. To find solutions of the initial equation in the form of the Laurent or Puiseux series we use algorithm of the Painlevé test.