Preliminary group classification of equations v t t=f (x,v x)v x x+g(x,v x)

A classification is given of equations vtt=f(v,vx)vxx+g(x,vx) admitting an extension by one of the principal Lie algebra of the equation under consideration. The paper is one of few applications of a new algebraic approach to the problem of group classification: the method of preliminary group classification. The result of the work is a wide class of equations summarized in Table II.