对于一类具有一条抛物线解、两条直线解和中心-焦点型奇点的三次系统,证明它以原点为中心的充要条件是它的第一阶焦点量为零. 系统在原点的中心条件是通过不变代数曲线构造积分因子或利用Poincar\'{e}对称原理得以证明.

A class of cubic systems with a parabola, two straight lines and a center-focus type singular point, are proved to have a center at the origin if and only if the first focal value vanishes. The center conditions of the system at the origin are proved by constructing integrating factors formed from invariant algebraic curves or by the Poincar\'{e} symmetry principle.