Constructing multiple prolongation structures from homotopic maps

In this paper, we show how multiple prolongation structures developed out of homotopy theory, can be constructed from a differential ideal corresponding to an exterior differential system. We use this method to construct multiple prolongation structures for the Robinson–Trautman equations of Petrov type III. It is found that the introduction of two arbitrary pseudo-potentials in the carrier space of the vector fields of this equation imposes nontrivial constraints on the prolongation structures which prevents the algebra from growing rapidly. Specific choices of the newly introduced pseudo-potentials result a coupled Kac–Moody A1⊕A1A1⊕A1 and Virasoro algebra as prolongation structure. Other choices of the potentials reproduce previously established results, namely the contragradient algebra K2K2 of infinite groiwth. The Lax pair and Riccati equations for pseudo-potentials can be formulated respectively from linear and nonlinear realizations of the prolongation structure.