$\mathbb{T}$-homogeneous locally nilpotent derivations of trinomial algebras

A trinomial algebra is a commutative finitely generated algebra given by a system of compatible relations each of which is a polynomial with three terms. Such algebras arise as the Cox rings of varieties admitting a complexity one torus action. We describe locally nilpotent derivations of a trinomial algebra that are homogeneous under a natural torus action of complexity one.