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Classification of solutions of a Toda system in $$\mathbb{R}^2$$. (English) Zbl 1001.35037
The paper deals with the 2-dimensional Toda system for SU(N+1) of the form $-\frac{1}{2}\triangle u_i=\sum \limits_{j=1}^Na_{ij}e^{u_j}\tag{1}$ in $${\mathbb R}^2$$ for $$i=1,2,\cdots ,N\in \mathbb N$$, where $$K=(a_{ij})_{N\times N}$$ is the Cartan matrix in SU(N+1). It is shown that any $$C^2$$-solution $$u=(u_1,u_2,\cdots ,u_N)$$ of (1) satisfying $$\int \limits_{{\mathbb R}^2}e^{u_i}<\infty$$, $$i=1,2,\dots ,N$$ has the form $u_i(z)=\sum \limits_{j=1}^Na_{ij}\log \|\Lambda_j(f)\|^2$ for some rational curve in $$\mathbb{C}\mathbb{P}^N$$. The analytic and geometric aspects of the Toda system are also presented.

##### MSC:
 35J60 Nonlinear elliptic equations 35Q53 KdV equations (Korteweg-de Vries equations)
##### Keywords:
Toda system; rational curve; holonomy
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