<p>In this paper, we consider the problem of a strongly nonlinear oscillator with movable algebraic singularities. The paper demonstrates the presence of such singularities and proves the existence and uniqueness theorem of the solution to the equation in the neighborhood of a moving singular point. An analytical approximate solution is constructed and the estimates of its error are obtained. Theoretical results were validated through numerical experiments. Furthermore, the problem of finding the point and interval criteria for the existence of the movable singular point was addressed. This problem allows for the development of an algorithm to find a moving singular point.</p>