  Preliminary group classification became prominent as an approach to symmetry
analysis of differential equations due to the paper by Ibragimov, Torrisi and
Valenti [J. Math. Phys. 32, 2988-2995] in which partial preliminary group
classification of a class of nonlinear wave equations was carried out via the
classification of one-dimensional Lie symmetry extensions related to a fixed
finite-dimensional subalgebra of the infinite-dimensional equivalence algebra
of the class under consideration. In the present paper we implement, up to both
usual and general point equivalence, the complete group classification of the
same class using the algebraic method of group classification. This includes
the complete preliminary group classification of the class and finding Lie
symmetry extensions which are not associated with subalgebras of the
equivalence algebra. The complete preliminary group classification is based on
listing all inequivalent subalgebras of the whole infinite-dimensional
equivalence algebra whose projections are qualified as maximal extensions of
the kernel algebra. The set of admissible point transformations of the class is
exhaustively described in terms of the partition of the class into normalized
subclasses. A version of the algebraic method for finding the complete
equivalence groups of a general class of differential equations is proposed.
