Lie symmetries and exact solutions of a new generalized Hirota–Satsuma coupled KdV system with variable coefficients

A new generalized Hirota–Satsuma coupled KdV system with variable coefficients is examined for Lie symmetry group and admissible forms of the coefficients with the help of the symmetry method based on the Fréchet derivative of the differential operators. An optimal system, of non-equivalent (non-conjugate) one dimensional sub-algebras of the symmetry algebra of the KdV system, having ten basic fields is determined. Using the non-equivalent Lie ansätze, for each essential vector field, the nonlinear system is reduced to systems of ordinary differential equations, and some special exact solutions of the KdV system are constructed.