  We work towards the classification of all one-dimensional exclusion processes
with two species of particles that can be solved by a nested coordinate Bethe
Ansatz. Using the Yang-Baxter equations, we obtain conditions on the model
parameters that ensure that the underlying system is integrable. Three classes
of integrable models are thus found. Of these, two classes are well known in
literature, but the third has not been studied until recently, and never in the
context of the Bethe ansatz. The Bethe equations are derived for the latter
model as well as for the associated dynamics encoding the large deviation of
the currents.
