A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems

"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group G are an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra 'S of the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of 'S vary in case of 3 < s < r, depending on the solvability of 'S. If the r-dimensional Lie algebra 'S of the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of 'S one has to determine the optimal subsystems of semisimple subalgebras of 'S in order to construct the full optimal systems of s-dimensional subalgebras of 'S with 3 < s <r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra 'S of the eight-dimensional Lie point symmetry group G admitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.