  It is shown that the first order (Palatini) variational principle for a
generic nonlinear metric-affine Lagrangian depending on the (symmetrized) Ricci
square invariant leads to an almost-product Einstein structure or to an
almost-complex anti-Hermitian Einstein structure on a manifold. It is proved
that a real anti-Hermitian metric on a complex manifold satisfies the K\"ahler
condition on the same manifold treated as a real manifold if and only if the
metric is the real part of a holomorphic metric. A characterisation of
anti-K\"ahler Einstein manifolds and almost-product Einstein manifolds is
obtained. Examples of such manifolds are considered.

Differential Geometry;General Relativity and Quantum Cosmology;High Energy Physics - Theory;Differential Geometry