  In this work, we employ the algebraic-differential method recently developed
by the author to solve the Yang-Baxter equation for arbitrary fifteen-vertex
models satisfying the ice-rule. We show that there are four different families
of such regular $R$ matrices containing several free-parameters. The
corresponding reflection $K$ matrices, solutions of the boundary Yang-Baxter
equation, were also found and classified. We found that there are three
different families of regular $K$ matrices, regardless of what $R$ matrix we
choose.
