Cosymmetries of chiral-type systems

We consider chiral-type systems admitting a Lax representation with values in a real or complex semisimple Lie algebra such that an additional regularity condition is satisfied (one of the matrices is a regular element of the Lie algebra). We prove that for a chiral-type system with vanishing torsion and a nonvanishing curvature, the existence of at least one pointwise cosymmetry is a necessary condition for the regular Lax representation.