In this paper, we primarily investigate Lie symmetry analysis and exact solutions for the coupled integrable dispersionless equations. First of all, based on the Lie symmetry analysis, an optimal system of one dimensional subalgebras is constructed. Furthermore, similarity reductions and group invariant solutions are given. Next, exact solutions of the reduced equations have been derived by the method of power series. Finally, by means of Ibragimov’s method, conservation laws are obtained.

In this paper, based on the classical symmetry method, the group-invariant solutions of the evolution equation of a hyperbolic curve flow are investigated. The optimal system of the obtained symmetries is found, and the reduced equations and exact solutions of the evolution equation are discussed. Then explicit solutions are obtained by the power series method. In addition, the convergence of the power series solutions is proved. Theobjective shapes of the solutions of the evolution equation are performed.