The study of limit cycles is an important topic in the qualitative theory of planar polynomial differential systems, since it helps to understand the periodic behavior of solutions and the global dynamics of nonlinear systems. In many situations arising in applied mathematics and dynamical systems, the existence, number, and distribution of limit cycles provide essential information about the long-term behavior of trajectories. However, obtaining explicit expressions for limit cycles is usually difficult, especially for higher-degree polynomial systems, where the analytical structure of periodic orbits becomes more complicated.

In this work, we study a planar polynomial differential system of degree nine. The system is constructed in such a way that it admits four explicit invariant algebraic curves surrounding the same singular point. These invariant curves are given in explicit form and play a fundamental role in the qualitative analysis of the system. Their presence allows us to investigate the structure of the periodic solutions and the configuration of closed orbits around the singular point.

First, we prove that the considered system possesses four invariant algebraic curves and determine their explicit expressions. Then, by applying a result reported by J. Giné and M. Grau, we show that each of these invariant curves corresponds to a limit cycle of the system. Consequently, the system admits four algebraic limit cycles surrounding the same singular point, and explicit expressions for these limit cycles are obtained.