  The halo dark matter (DM) can be gravitationally captured by the Sun. For
self-interacting DM (SIDM), we show that the number of DM trapped inside the
Sun remains unsuppressed even if the DM-nucleon cross section is negligible. We
consider a SIDM model where $U(1)$ gauge symmetry is introduced to account for
the DM self-interaction. Such a model naturally leads to isospin violation for
DM-nucleon interaction, although isospin symmetry is still allowed as a special
case. We show that the detection of neutrino signature from DM annihilation in
the Sun can probe those SIDM parameter ranges not reachable by direct
detections. Those parameter ranges are either the region with a very small
$m_{\chi}$ or the region opened up due to isospin violations.
