  We study the quantum moduli space of vacua of $N=2$ supersymmetric $SU(N_c)$
gauge theories coupled to $N_f$ flavors of quarks in the fundamental
representation. We identify the moduli space of the $N_c = 3$ and $N_f=2$
massless case with the full spectral curve obtained from the Lax representation
of the Goryachev-Chaplygin top. For the case with {\it massive} quarks, we
present an integrable system where the corresponding hyperelliptic curve
parametrizing the Laurent solution coincides with that of the moduli space of
$N_{c}=3$ with $N_{f}=0, 1, 2$. We discuss possible generalizations of the
integrable systems relevant to gauge theories with $N_c \neq 3 $ and general
$N_f$.
