Koenigs problem and extreme fixed points

This note continues some previous studies by the authors. We consider a linear-fractional mapping $ F_A :K to K $ F_A :K to K generated by a triangular operator, where $ K $ K is the unit operator ball and the fixed point C of the extension of $ F_A $ F_A to $ overline K $ overline K is either an isometry or a coisometry. Under some natural restrictions on one of the diagonal entries of the operator matrix A, the structure of the other diagonal entry is investigated completely. It is shown that generally C cannot be replaced in all these considerations by an arbitrary point of the unit sphere. Some special cases are studied in which this is nevertheless possible.