Exact solutions of the transonic equations of gas dynamics

The review exact (described by algebraic functions) solutions of a transonic set of Karman–Falkovitch equations is given. Self-similar solutions and two classes of the polynomial-parametrical solutions associated with self-similar at indexes $n=2$ and $n=3$ are considered. Connection with local exposition of singularities of transonic flows is specified, in particular in Laval nozzles.