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It is established that the existence of non-isotropic vector field whichJacobi operator of maximal rank is an obstacle for the existence of non-trivialsecond-order symmetric parallel tensor field. In turns out that presence ofsuch obstacle follows that manifold as pseudo-Riemannian manifold is locallynon-reducible. In particular result can be applied directly to known classes ofalmost (para-) contact metric manifolds when considered Jacobi operator ofcharacteristic vector field has maximal rank. There is effective algorithmicprocedure which resolves the problem of existence of such vector field in purealgebraic 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 andonly if it is non-zero of this form.

Rank of Jacobi operator and existence of quadratic parallel differential form, with applications to geometry of almost para-contact metric manifolds