  In a previous paper, we showed that the problem of tunneling in quantum wires
was integrable in the isotropic case $g_\sigma=2$. In the present work, we
continue the exploration of the general phase diagram by looking for other
integrable cases. Specifically, we discuss in details the manifold
$g_\rho+g_\sigma=2$, where the associated ``double sine-Gordon'' model is
integrable. Transport properties are exactly computed. Surprisingly, the IR
fixed points, while having complete reflection of charge and spin currents, do
not correspond to two separate leads. Their main characteristic is that they
are approached along irrelevant operators of dimension $1+{1\over g_\rho}$ and
$1+{1\over g_\sigma}$, corresponding to transfer of one electron charge but no
spin, or one spin 1/2 but no charge.
