This study explores the infinite group structures related to integrabilities of a solvable wave equation proposed by Calogero. We find the group splittings as well as the linearization mapping of the equation and its equivalent system. The equivalent system is found to be an automorphic system with respect to an infinite group. It can also be split into an automorphic system and a resolving system which can be solved by quadratures. In English literature a concrete example is difficult to find that illustrates the notion of reducing a non-linear PDE with order higher than one to quadratures by the method of group splitting. Our results indicate that the equivalent system can be served as a good example in this aspect. The results obtained also provide a group-theoretic interpretation of the solvability of the equation, which had not been completely developed in Calogero's original work. Since the equation does not pass the Painleve test, our results demonstrate that sometimes group analysis can obtain much more information than the Painleve test does in detecting the integrabilities of non-linear PDEs. (C) 1997 Elsevier Science Ltd.
