Trigonal Toda lattice Equation

In this article, we give the trigonal Toda lattice equation, $-\frac{1}{2}\frac{d^3}{d t^3} q_{\ell}(t) = e^{q_{\ell+1}(t)} +e^{q_{\ell+ζ_3}(t)} +e^{q_{\ell+ζ_3^2}(t)}-3e^{q_\ell(t)},$ for a lattice point $\ell \in \mathbb{Z}[ζ_3]$ as a directed 6-regular graph where $ζ_3=e^{2πi/3}$, and its elliptic solution for the curve $y(y-s)=x^3$, ($s\neq 0$).

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