We study a modified version of an equation of the continuous Toda type in 1 + 1 dimensions. This equation contains a friction-like term which can be switched off by annihilating a free parameter . We apply the prolongation method, and the symmetry and approximate symmetry approaches. This strategy allows us to gain insight into both the equations for and , whose properties arising from the above frameworks we compare. For , the related prolongation equations are solved by means of certain series expansions which lead to an infinite-dimensional Lie algebra. Furthermore, using a realization of the Lie algebra of the Euclidean group , a connection is shown between the continuous Toda equation and a linear wave equation which resembles a special case of a three-dimensional wave equation that occurs in a generalized Gibbons - Hawking ansatz (Lebrun C 1991 J. Diff. Geom. 34 223). Non-trivial solutions to the wave equation expressed in terms of Bessel functions are determined.

For , we obtain a finite-dimensional Lie algebra with four elements. A matrix representation of this algebra yields solutions of the modified continuous Toda equation associated with a reduced form of a perturbative Liouville equation. This result coincides with that achieved in the context of the approximate symmetry approach. Example of exact solutions are also provided. In particular, the inverse of the exponential-integral function turns out to be defined by the reduced differential equation arising from a linear combination of the time and space translations. Finally, a Lie algebra characterizing the approximate symmetries is discussed.