Outer automorphisms of $sl(2)$, integrable systems, and mappings

Outer automorphisms of infinite-dimensional representations of the Lie algebra $sl(2)$ are used to construct Lax matrices for integrable Hamiltonian systems and discrete integrable mappings. The known results are reproduced, and new integrable systems are constructed. Classical $r$-matrices corresponding to the Lax representation with the spectral parameter are dynamic. This scheme is advantageous because quantum systems naturally arise in the framework of the classical $r$-matrix Lax representation and the corresponding quantum mechanical problem admits a variable separation.