The continuous dependence of solutions to algebraic differential systems on the initial data

We consider a nonlinear system of ordinary differential equations, which is unsolved with respect to the derivative of the desired vector function and identically degenerate in the definition domain. We study the consistency manifold under assumptions that guarantee the existence of a solution. We prove an analog of the theorem on the continuous dependence of solutions on the initial data, assuming that the latter belong to the consistency manifold.