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

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