In this paper we deal with the Liénard system \(\dot{x}=y, \dot{y}=-f_m(x)y-g_n(x),\) where \(f_m(x)\) and \(g_n(x)\) are real polynomials of degree m and n, respectively. We call this system the Liénard system of type (m, n). For this system, we proved that if \(m+1\le n\le [\frac{4m+2}{3}]\), then the maximum number of hyperelliptic limit cycles is \(n-m-1\), and this bound is sharp. This result indicates that the Liénard system of type \((m,m+1)\) has no hyperelliptic limit cycles. Secondly, we present examples of irreducible algebraic curves of arbitrary high degree for Liénard systems of type \((m,2m+1)\). Moreover, these systems have a rational first integral. Finally, we proved that the Liénard system of type (2, 5) has at most one hyperelliptic limit cycle, and this bound is sharp.

在本文中，我们处理Liénard系统\（\ dot {x} = y，\ dot {y} =-f_m（x）y-g_n（x），\）其中\（f_m（x）\）和\ （g_n（x）\）分别是阶次为m和n的实多项式。我们称这个系统为（m，  n）类型的Liénard系统。对于该系统，我们证明了如果\（m + 1 \ le n \ le [\ frac {4m + 2} {3}] \），则超椭圆极限环的最大数量为\（nm-1 \），这个界限是尖锐的。该结果表明\（（m，m + 1）\）类型的Liénard系统没有超椭圆极限环。其次，我们给出\（（m，2m + 1）\）类型的Liénard系统的任意高阶不可约代数曲线的示例。而且，这些系统具有合理的第一积分。最后，我们证明了类型（2，5）的Liénard系统最多具有一个超椭圆极限环，并且该边界是尖锐的。