MY AIM IN THIS TALK IS TO DIVIDE THE SET OF ALL INVARIANT
SOLUTIONS OF A GIVEN DIFFERENTIAL EQUATION INTO
EQUIVALENCE CLASSES. TWO INVARIANT SOLUTIONS ARE
EQUIVALENT IF ONE CAN BE MAPPED TO THE OTHER BY A POINT
SYMMETRY OF THE DIFFERENTIAL EQUATION. CLASSIFICATION
GREATLY SIMPLIFIES THE PROBLEM OF DETERMINING ALL
INVARIANT SOLUTIONS. IT IS ONLY NECESSARY TO FIND ONE
GENERAL INVARIANT SOLUTION FROM EACH CLASS; THEN THE
WHOLE CLASS CAN BE CONSTRUCTED BY APPLYING THE
SYMMETRIES. THIS STRATEGY MINIMIZES THE EFFORT NEEDED
TO OBTAIN INVARIANT SOLUTIONS. WE DISCUSS THE ONE-
DIMENSIONAL OPTIMAL SYSTEM OF SUB-ALGEBRAS FOR A WAVE
EQUATION WITH ARBITRARY FUNCTION. IN PARTICULAR, I WILL
START WITH THE PRINCIPAL LIE ALGEBRA AND CONSTRUCT A
SET OF SYMMETRY TRANSFORMATIONS THAT FORM A ONE-
DIMENSIONAL OPTIMAL SYSTEM FOR THE GIVEN EQUATION.