Invariant and partially invariant solutions with respect to Galilean shifts and dilatation

In the work we consider a three-dimensional subalgebra embedded in a four-dimensional subalgebra in order to find the set of solutions and to adjoint them the solutions on subalgebras of higher dimension. Although the aim is not reached yet, we obtain invariant solutions of the rank 1 and partially invariant solutions of the rank 1 and defect 1. We obtain two submodels being invariant and partially invariant, seven solutions depend on arbitrary function and nineteen exact solutions.