For the polynomial differential system \(\begin{aligned} \dot{x}=-y+\sum \limits _{i+j=3}\alpha _{i,j}x^iy^j,\ \dot{y}=x+\sum \limits _{i+j=3}\beta _{i,j}x^iy^j,\ \alpha _{i,j},\beta _{i,j}\in {\mathbb {R}}, \end{aligned}\) Pleshkan (Differ. Equations, 1969) established that the origin is an isochronous center of this system if and only if it can be transformed into one of the canonical forms \(S^*_1\) , \(S^*_2\) , \(S^*_3\) or \(S^*_4\) . Except for case \(S^*_1\) , the bifurcations of limit cycles in these four types of isochronous differential systems remain unexplored. In this paper, we focus on the bifurcation of limit cycles for the system \(S^*_2\) under perturbations by an arbitrary polynomial vector field. By employing the Abelian integral, we derive an upper bound for the number of limit cycles that can emerge from such perturbations. The lower bounds are also provided for \(n=1, 2, 3, 4\) , and numerical simulations are conducted.
