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We study limiting cases of the two known integrable chiral-type models withtree-dimensional configuration space. One of the initial models is thenon-Abelian Toda $A_2^{(1)}$ model and the other was found by means of thesymmetry approach by A.G. Meshkov and one of the authors. The C-integrabilityof the reduced models is established by constructing their complete sets ofintegrals and general solutions. A description of the generalized symmetryalgebras of these models is given in terms of operators mapping integrals intosymmetries. The integrals of the Liouville-type systems are known to defineMiura-type transformations for their generalized symmetries. This fact allowedus to find a few new systems of the Yajima-Oikawa type. We present a recursionoperator for one them.

On non-Abelian Toda $A_2^{(1)}$ model and related hierarchies