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 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 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.