Second-order generalized Monge--Amp\`ere equations on a plane and its geometric singular solutions

In the present paper, we study some generalized Monge–Amp`ere equations in terms of special exterior differential systems on a jet space. Moreover, we construct geometric singular solutions of the generalized Monge–Amp`ere equations by using the method of Cauchy characteristics. Furthermore, we give criteria for geometric singular solutions to be right-left equivalent to the cuspidal edge, swallowtail and butterfly.