A theorem on two commuting automorphisms, and integrable differential equations

Constructions are found for differential equations in an arbitrary continuous associative algebra $\mathfrak A$ that admit an equivalent Lax representation (with spectral parameter) in the space of linear operators acting on $\mathfrak A$. The constructions use commuting automorphisms of $\mathfrak A$. Applications of the main construction are indicated for the construction of integrable Euler equations in the direct sum of the Lie algebras $\operatorname{gl}(n,R)$ and $\operatorname{so}(n,R)$. Constructions are presented for matrix differential equations admitting a Lax representation with several spectral parameters.