It is shown that for a two-dimensional connected homogeneous Kählerian manifold the canonical Hermite form of which is degenerate but non-zero the canonical Hano-Kobayashi fibering is holomorphic. Because the fibers of this fibering are homogeneous Kählerian manifolds with zero Ricci curvature they turn out to be locally flat Kählerian manifolds. Likewise it is established in the two-dimensional case the validity of the conjecture of the construction of homogeneous Kählerian manifolds belonging to Gindikin, Pjatetskij-Shapiro and the reviewer. Another conclusion of the obtained result is that a two-dimensional homogeneous Kählerian manifold containing no complex straight line turns out to be a homogeneous bounded domain. In the paper it is used the technique of Kähler algebras.