A classification is given of equations v_tt=f(v,v_x)v_xx+g(x,v_x) 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.

A simple procedure is suggested to obtain extensions of the principal Lie algebra, for invariance transformations of a given family of equations, by using an equivalence algebra. An application to a qualitative model of detonation is performed.