On the structure of two-dimensional local skew fields

The concept of $n$-dimensional local skew field is a direct generalization of the concept of $n$-dimensional local field. We study 2-dimensional local skew fields and solve the classification problem for the those of characteristic 0 whose last residue field is contained in the centre, and suggest a condition under which there is a section of the residue map whose first residue skew field is commutative. Under this condition we solve the classification problem for all 2-dimensional local skew fields.
For skew fields of characteristic 0 whose last residue field is contained in the centre, we state a criterion for two elements to be conjugate.