Classification of solutions of a Toda system in ℝ2

We consider solutions of the following (open) Toda system (Toda lattice) for SU(N + 1)$-\frac12\Delta u_i=\sum_{j=1}^Na_{ij}\text{?}e^{u_j}\text{???in?}\mathbb{R}^2\qquad(1.1)$, for i = 1, 2, …, where $K=(a_{ij})_{N\times N}$ is the Cartan matrix for SU(N + 1). We show that any solution u = (u1,u2,…,uN) with$\int_{\mathbb{R}^2}e^{u_i}<\infty,\text{??????}i=1,2,{\cdots},N$, can be obtained from a rational curve in ℂP^N.