  In this present paper, we study geometric structures of rank two
prolongations of implicit second-order partial differential equations (PDEs)
for two independent and one dependent variables and characterize the type of
these PDEs by the topology of fibers of the rank two prolongations. Moreover,
by using properties of these prolongations, we give explicit expressions of
geometric singular solutions of second-order PDEs from the point of view of
contact geometry of second order.
