  We apply an extension of a new method of group classification to a family of
nonlinear wave equations labelled by two arbitrary functions, each depending on
its own argument. The results obtained confirm the efficiency of the proposed
method for group classification, termed the method of indeterminates. A model
equation from the classified family of fourth order Lagrange equations is
singled out. Travelling wave solutions of the latter are found through a
similarity reduction by variational symmetry operators, followed by a double
order reduction into a second order ordinary differential equation.
Multi-soliton solutions and other exact solutions are also found by various
methods including Lie group and Hirota methods. The most general action of the
full symmetry group on any given solution is provided. Some remarkable facts on
Lagrange equations emerging from the whole study are outlined.
