On the Goursat classification problem

In the paper, the Goursat problem—classification of nonlinear hyperbolic differential equations possessing two characteristic invariants—is considered. An algorithm for finding the characteristic invariants is described. On the basis of the algorithm implemented in REDUCE, the characteristic invariants of two Laine’s equations are verified. One of them is shown to have invariants of the second and third orders. This equation shows that the Goursat problem, apparently, is still open. Computer calculations show that the characteristic invariants of the second Laine’s equation given in his paper are incorrect.