Reflection K-matrices for 19-vertex models

We derive and classify all regular solutions of the boundary Yang-Baxter equation for 19-vertex models known as Zamolodchikov-Fateev or A₁⁽¹⁾ model, Izergin-Korepin or A₂⁽²⁾ model, sl(2|1) model and the osp(2|1) model. We find that there is a general solution for A₁⁽¹⁰⁾ and sl(2|1) models. In both models it is a complete K-matrix with three free parameters. For the A₂⁽²⁾ and os(2|1) models we find three general solutions, being two complete reflection K-matrices solutions and one incomplete reflection K-matrix solution with some null entries. In both models these solutions have two free parameters. Integrable spin-1 Hamiltonians with general boundary interactions are also presented. Several reduced solutions from these general solutions are presented in the appendices.