  There are five six-dimensional nilpotent Lie groups G, which do not admit
neither symplectic, nor complex structures and, therefore, can be neither
almost pseudo-Kahler, nor almost Hermitian. In this work, these Lie groups are
being studied. The aim of the paper is to define new left-invariant geometric
structures on the Lie groups under consideration that compensate, in some
sense, the absence of symplectic and complex structures. New examples of
multiparametric families of metrics of signature (3,3) and almost para-complex
pseudo-Riemannian half-flat structures on six-dimensional nilmanifolds are
obtained. These metrics have a diagonal Ricci operator with two eigenvalues,
which differ only in sign.
