Normal form of a binary polynomial in the critical point of the second order

We consider a real polynomial of two variables. Its expansion in the vicinity of the zero singular point begins with the third degree form. We find its simplest forms to which this polynomial is reduced by reversible real local analytic coordinate substitutions. First, the normal forms for the cubic form are obtained using linear coordinate substitutions. There are three of them. Then three non-linear normal forms were obtained for the full polynomial. A simplification of the computation of the normal form is proposed. A meaningful example is considered.