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Field equations for generalized principal chiral models with non-constant metric and their possible Lax formulation are considered. Ansatz for Lax operators is taken linear in currents. Results of a complete investigation of models allowing Lax formulation with linear ansatz for Lax operators on solvable 2- and 3-dimensional groups are given; all such models appear to be almost linear. Also models on simple group SU(2) with diagonal metric are considered; it turns out that Lax formulation exists in this case for constant metrics only.