  We apply the method of spectral sequences to study classical problems in
analysis. We illustrate the method by finding polynomial vector fields that
commute with a given polynomial vector field and finding integrals of
polynomial Hamiltonian systems. For the later we describe the integrals for the
Henon-Heiles Hamiltonian which arises in celestial mechanics. The unifying
feature is that these problems seek elements in the kernel of a linear
operator. The spectral sequence approach emphasizes the obstructions
constructed from cokernel of the operator to finding elements in the kernel.
