Integrability conditions for the equations of an asymmetric chiral field on $SO(4)$

The possibility of integrating the equations of an asymmetric chiral field on $SO(4)$ by the inverse scattering method is investigated. The maximal number of integrals of the motion of the corresponding system of ordinary differential equations is found. It is shown that when the condition (27) is satisfied, the system cannot be integrated by this method except for the cases (28) and (29).