On approximately integrable $SO(3)$ structures on 5-dimensional manifolds

In this work, irreducible $SO(3)$ structures on a 5-dimentional manifold are considered. The covariant divergence of the structure tensor is shown to be zero for approximately integrable irreducible $SO(3)$ structures. Examples of left invariant irreducible $SO(3)$ structures on 5-dimentional Lie groups that have a zero covariant divergence of the structure tensor but are not approximately integrable, as well as of irreducible $SO(3)$ structures with nonzero covariant divergence of the structure tensor are presented.