On conservation laws and zero-curvature representations of the Liouville equation

Abstract

Applying the first Noether theorem to the Liouville equation uxy=exp u, we find all (namely, a continuum of) non-trivial conservation laws of this equation. Then we find five new zero-curvature representations of the Liouville equation (by 2*2 traceless matrices) which contain, respectively, 1, 1, 2, 2 and 3 essential parameters. Finally, we show that all known zero-curvature representations of the Liouville equation are equivalent (in a definite sense) to matrices of conservation laws.