Motivated by recent work on frustrated spin chains and anyonic fusion chains, we define a 1D quantum Hamiltonian (based on the Temperley-Lieb algebra) which contains these models as special cases. This Hamiltonian contains a new integrable point with Z_2-symmetry, which we have solved exactly. In this talk, I will first explain the physical motivations and recall basic facts on the Temperley-Lieb algebra, a central object in 2D Statistical Mechanics. Then I will present analytical and numerical results for the phase diagram, and comment on perspectives in the fields of Condensed Matter Theory and Integrable Models.