Reducible Abelian varieties and Lax matrices for Euler’s problem of two fixed centres

Abel's quadratures for integrable Hamiltonian systems are defined up to a group law of the corresponding Abelian variety $A$. If $A$ is isogenous direct product varieties $A\cong A_1\times\cdots\times A_k$, can be used construct various Lax matrices on factors $A_1,\ldots,A_k$. As an example, we discuss 2-dimensional reducible $A=E_+\times E_-$, which 1-dimensional $E_\pm$ obtained by Euler in his study two fixed centres problem, and $E_\pm$.