  The Boussinesq $abcd$ system arises in the modeling of long wave small
amplitude water waves in a channel, where the four parameters $(a,b,c,d)$
satisfy one constraint. In this paper we focus on the solitary wave solutions
to such a system. In particular we work in two parameter regimes where the
system does not admit a Hamiltonian structure (corresponding to $b \ne d$). We
prove via analytic global bifurcation techniques the existence of solitary
waves in such parameter regimes. Some qualitative properties of the solutions
are also derived, from which sharp results can be obtained for the global
solution curves. Specifically, we first construct solutions bifurcating from
the stationary waves, and obtain a global continuous curve of solutions that
exhibits a loss of ellipticity in the limit. The second family of solutions
bifurcate from the classical Boussinesq supercritical waves. We show that the
curve associated to the second class either undergoes a loss of ellipticity in
the limit or becomes arbitrarily close to having a stagnation point.
