Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case

We obtain a parametric normal form for any non-degenerate perturbation of the generalized saddle-node case of Bogdanov–Takens singularity. Explicit formulas are derived and greatly simplified for an efficient implementation in any computer algebra system. A Maple program is prepared for an automatic parametric normal form computation. A section is devoted to present some practical formulas which avoid technical details of the paper.