Classification of Solutions with Polynomial Energy Growth for the SU (n + 1) Toda System on the Punctured Complex Plane

This paper investigates the classification of solutions satisfying the polynomial energy growth condition near both the origin and infinity to the ${\mathrm SU}(n+1)$ Toda system on the punctured complex plane $\mathbb{C}^*$. The ${\mathrm SU}(n+1)$ Toda system is a class of nonlinear elliptic partial differential equations of second order with significant implications in integrable systems, quantum field theory, and differential geometry. Building on the work of A. Eremenko (J. Math. Phys. Anal. Geom., Volume 3 p.39-46), Jingyu Mu's thesis, and others, we obtain the classification of such solutions by leveraging techniques from the Nevanlinna theory. In particular, we prove that the unitary curve corresponding to a solution with polynomial energy growth to the ${\mathrm SU}(n+1)$ Toda system on $\mathbb{C}^*$ gives a set of fundamental solutions to a linear homogeneous ODE of $(n+1)^{th}$ order, and each coefficient of the ODE can be written as a sum of a polynomial in $z$ and another one in $\frac{1}{z}$.