  The main focus of the paper is the investigation of moduli space of left
invariant pseudoRiemannian metrics on the cotangent bundle of Heisenberg group.
Consideration of orbits of the automorphism group naturally acting on the space
of the left invariant metrics allows us to use the algebraic approach. However,
the geometrical tools, such as classification of hyperbolic plane conics, will
often be required. For metrics that we obtain in the classification, we
investigate geometrical properties: curvature, Ricci tensor, sectional
curvature, holonomy and parallel vector fields. The classification of algebraic
Ricci solitons is also presented, as well as classification of pseudo-Kahler
and ppwave metrics. We get the description of parallel symmetric tensors for
each metric and showthat they are derived from parallel vector fields. Finally,
we investigate the totally geodesic subalgebras by showing that for any
subalgebra of the observed algebra there exists a metric that makes it totally
geodesic.
