Abstract
where ck(x), ψk(x) are unknown functions of the scalar variable x. They have been called the vector analogs of the Cauchy equation. Note that the classic Cauchy equation is a particular case of (1) corresponding to N = 0, and in this case all solutions of (1) are exponential functions. It was proved that for N = 1 and N = 2 all the solutions of this equation are the Baker-Akhiezer functions corresponding to algebraic curves of genus 1 and 2, respectively. A starting point for the consideration of (1) in [5] was close connections of this equation with the theory of one-dimensional integrable systems of the Calogero-Moser system type. The main goal of this note is to show that the multivariable Baker-Akhiezer functions give solutions of the following multivariable generalization of (1):
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