$K$-spaces of maximal rank

We consider a special type of $K$-space, i.e., almost-Hermitian manifolds whose fundamental form is a Killing form. The $K$-spaces of this type are characterized by the fact that their dimension is equal to the rank of the covariant derivative of the structure form. A number of the properties of such spaces are established (they are Einstein, compact, have finite fundamental group, etc.). It is proved that every $K$-space is locally equivalent to a product of a $K$-space of maximal rank and a Kähler manifold. The $K$-spaces with zero holomorphic sectional curvature are studied.