  We have investigated the phase dynamics and IV-characteristics of shunted
Josephson junctions coupled to an $LC$ circuit. When the Josephson frequency
$\omega_J$ is close to the eigen frequency $\omega_{rc}$ of the coupled
resonance circuit, the IV-characteristic demonstrates an additional
$rc$-branch. We have investigated the features of the $rc$-branch and of the
superconducting current component for different values of the resonance
frequency. It has been found that the maximal value of the superconducting
current depends on the resonance frequency and that it increases when the end
point of the $rc$-branch approaches the critical current. The dependence of the
maximal superconducting current on the resonance frequency at different values
of the dissipation parameter is peculiar, for the optimized maximum appears to
be independent of the system parameters within $1\%$.
