Maple Encoding of Optimal Systems for Solvable Lie Algebras

This paper presents a Maple code for the systematic computation of optimal systems of solvable Lie algebras, built on an ideal based algorithmic framework. The main contribution is a fully automated Maple implementation that computes ideals, possible representative forms, adjoint factorizations, and conjugacy classes in a single reproducible workflow. The code is validated on a 9-dimensional solvable Lie algebra, where all ideals were determined in approximately five minutes, while the subsequent forms, adjoint factorizations, and conjugacy classes were each obtained in under a minute, a task that would take considerably longer manually. One-dimensional representatives are further extended to two-dimensional subalgebras through normalizers and quotient algebras. The code is then applied to the Lax equation, where its optimal system is used to derive invariant solutions, which are analyzed graphically to illustrate their behavior in detail.