An ansatz for elliptic Baker-Akhiezer functions is used to find new solutions for various spectral problems--the Sturm-Liouville problem with four and five-gap Lamé potentials and two third-order problems. One of the third-order problems is the classical Halphen equation with a spectral parameter. The eigenfunctions are used to construct elliptic solutions for integrable cases of a generalized Hénon-Heiles system.
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