  It is established that the existence of non-isotropic vector field which
Jacobi operator of maximal rank is an obstacle for the existence of non-trivial
second-order symmetric parallel tensor field. In turns out that presence of
such obstacle follows that manifold as pseudo-Riemannian manifold is locally
non-reducible. In particular result can be applied directly to known classes of
almost (para-) contact metric manifolds when considered Jacobi operator of
characteristic vector field has maximal rank. There is effective algorithmic
procedure which resolves the problem of existence of such vector field in pure
algebraic way - there is canonically defined homogeneous differential form,
with coefficients determined purely by the coeffcients of curvature operator -
such that non-isotropic vector field has non-degenerate Jacobi operator if and
only if it is non-zero of this form.
