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The basic idea is, that such a system can be written as a Lax pair: $$\dot{L} = [P_j, L]$$ for some $j$. Different $j$ correspond to different members of the hierachy. Now to construct such an algebro-geometric solution. Consider some $\ell > j$, and look for an $L$ such that $$[L, P_{\ell}] = 0$$ then some general theory implies that this solution satisfies a polynomial equation, and can thus be written in terms of data on a curve.