A Note of Description on the Lie Algebra of the Invariants in the CBS Nonlinear Equation

In this short note, a particular realization of the vector fields that form a Lie Algebra of symmetries for
the Calogero-Bogoyavleskii-Schiff equation is found. The Lie Algebra is examined and the result is a
semidirect product of two Lie Groups. The structure of the semidirect product is examined through
the table of commutation rules. Two reductions are made with the help of two sets of generators and
the final outcome for the solution is related to the elliptic Painlevé -function.