Summary: As is well known, there are \(34\) classes of isomorphic simply connected six-dimensional nilpotent Lie groups. Of these, only \(26\) classes admit left-invariant symplectic structures and only \(18\) classes admit left-invariant complex structures. There exist five six-dimensional nilpotent Lie groups \(G\), which do not admit neither symplectic, nor complex structures and, therefore, can be neither almost pseudo-Kählerian, nor Hermitian. It is the Lie groups that are studied in this work. The aim of the paper is to define new left-invariant geometric structures on the Lie groups. If the left-invariant \(2\)-form \(\omega\) on such a Lie group is closed, then it is degenerate. Weakening the closedness requirement for left-invariant \(2\)-forms \(\omega \), stable \(2\)-forms \(\omega\) are obtained. Their exterior differential \(d\omega\) is also stable in Hitchin sense. Therefore, the pair \((\omega, d\omega)\) defines either an almost Hermitian or almost para-Hermitian structure on the group \(G\). The corresponding pseudo-Riemannian metrics are Einstein for four of the five Lie groups under consideration. This gives new examples of multiparameter families of left-invariant Einstein pseudo-Riemannian metrics on six-dimensional nilmanifolds. On each of the Lie groups under consideration, compatible and normalized pairs of left-invariant forms \((\omega,\rho)\), where \(\rho=d\omega \), are obtained. They define semi-flat structures. The Hitchin flow on \(G\times I\) is studied to construct a pseudo-Riemannian metric on \(G\times I\) with a holonomy group from \(G_2^*\) and it is shown that there is nots solution in this class of left-invariant half-plane structures \((\omega,\rho)\). For structures \((\omega,\rho)\), only the \(3\)-form closure property \(\varphi=\omega \wedge dt+d\omega\) on \(G\times I\) holds.