The first integrals and rational solutions of differential equations with a moving singular line

The nonlinear autonomous higher-order differential equations with a moving singular line are studied. The first integrals of the Chazy equation and some other equations with a moving singular line are obtained. Nonlinear differential equations, for which general solutions are rational solutions of equations with a moving singular line are obtained. It is shown that with the help of Bäcklund transformations, rational solutions of equations with a movable singular line can be transformed into each other. Nonlinear differential equations of the second or third degree with respect to the highest derivative are obtained, for which these rational solutions are general solutions.