Lie-algebraic K\"ahler sigma models with the U(1) isotropy

We discuss various questions which emerge in connection with the Lie-algebraic deformation of CP¹ sigma model in two dimensions. First we supersymmetrize the original model endowing it with the minimal N=(0,2) and extended N=(2,2) supersymmetries. Then we derive the general hypercurrent anomaly in the both cases. In the latter case this anomaly is one-loop but is somewhat different from the standard expressions one can find in the literature because the target manifold is non-symmetric. We also show how to introduce the twisted masses and the θ term, and study the BPS equation for instantons, in particular the value of the topological charge. Then we demonstrate that the second loop in the β function of the non-supersymmetric Lie-algebraic sigma model is due to an infrared effect. To this end we use a supersymmetric regularization. We also conjecture that the above statement is valid for higher loops too, similar to the parallel phenomenon in four-dimensional N=1 super-Yang-Mills. In the second part of the paper we develop a special dimensional reduction -- namely, starting from the two-dimensional Lie-algebraic model we arrive at a quasi-exactly solvable quantum-mechanical problem of the Lam\'e type.