The a(2)_4 Toda lattice is claimed to linearize on the Jacobian of an explicit genus-2 curve and to admit a Lax pair, but the Lax pair as stated contradicts the system's equations.
and Birkenhake, C., Abelian Varieties over the Complex Numbers, A Graduate Course, Grundlehren Text Editions
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Linearization, separability and Lax pairs representation of $a_4^{(2)}$ Toda lattice
The a(2)_4 Toda lattice is claimed to linearize on the Jacobian of an explicit genus-2 curve and to admit a Lax pair, but the Lax pair as stated contradicts the system's equations.