  We study 2D non-linear sigma models on a group manifold with a special form
of the metric. We address the question of integrability for this special class
of sigma models. We derive two algebraic conditions for the metric on the group
manifold. Each solution of these conditions defines an integrable model.
Although the algebraic system is overdetermined in general, we give two
examples of solutions. We find the Lax field for these models and calculate
their Poisson brackets. We also obtain the renormalization group (RG)
equations, to first order, for the generic model. We solve the RG equations for
the examples we have and show that they are integrable along the RG flow.
