  We show that the most general inflaton potential in Minkowski spacetime for
which the differential equation for the Fourier modes of the matter fields
reduces to Lam\'e's equation is of the form $V(\phi)=
\lambda\phi^4/4+K\phi^2/2+\mu/(2\phi^2)+V_0$. As an application, we study the
preheating phase after inflation for the above potential with K=0 and arbitrary
$\lambda,\mu>0$. For certain values of the coupling constant between the
inflaton and the matter fields, we calculate the instability intervals and the
characteristic exponents in closed form.
