Trigonal Toda lattice Equation

In this article, we give the trigonal Toda lattice equation, $$-\frac{1}{2}\frac{d^3}{d t^3} q_{\ell}(t) = e^{q_{\ell+1}(t)} +e^{q_{\ell+\zeta_3}(t)} +e^{q_{\ell+\zeta_3^2}(t)}-3e^{q_\ell(t)},$$ for a lattice point $\ell \in \mathbb{Z}[\zeta_3]$ as a directed 6-regular graph where $\zeta_3=e^{2\pi i/3}$, and its elliptic solution for the curve $y(y-s)=x^3$, ($s\neq 0$).