Tensor fields associated with integrable systems of chiral type

This article describes necessary conditions for chiral-type systems to admit Lax representation with values in simple compact Lie algebras. These conditions state that there exists a covariant constant tensor field with an additional property. It is proposed to construct in an invariant way some covariant tensor fields using the Lax representation of the system under consideration. These fields are constructed by taking linear differential forms with values in the Lie algebra that are constructed using the Lax representation of the system and substituting them into an arbitrary Ad-invariant form on the Lie algebra. The paper proves that such tensor fields are Killing tensor fields or covariant constant fields. The discovered necessary conditions for the existence of the Lax representation are obtained using a special case of such tensor fields associated with the Killing metric of the Lie algebra.