Abstract
In this article, we give the trigonal Toda lattice equation,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - {1 \\over 2}{{{d^3}} \\over {d{t^3}}}q\\ell (t)\\, = \\,{e^{q\\ell + 1(t)}}\\, + \\,{{\\rm{e}}^{q\\ell + {\\zeta _3}^{(t)}}}\\, + \\,{{\\rm{e}}^{{q_\\ell } + \\,\\zeta {{_3^2}^{(t)}}}}\\, - \\,3{{\\rm{e}}^{q\\ell {\\rm{(}}t{\\rm{)}}}}$$\\end{document}for a lattice point £ ∊ ℤ[ζ3] as a directed 6-regular graph where \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\zeta _3}\\, = \\,{{\\rm{e}}^{{\\rm{2\\pi }}\\sqrt { - 1/3} }}$$\\end{document}, and its elliptic solution for the curve y(y − s) = x3, (s ≠ 0).
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