Fouling maps and polynomial first integrals from symmetric tensor fields

Under the framework of time-independent Hamiltonian mechanics on the cotangent bundles $T^*Q$ of the configuration spaces $Q$ of mechanical systems, we introduce the concept of fouling map as a non-invertible generalization of the so-called fouling transformations --canonoid transformations preserving configuration coordinates--. We develop a tensorial method for constructing polynomial fouling maps. We show that each such map induces a $(1,1)$-tensor field invariant under the Hamiltonian flow, whose traces of its powers are polynomial constants of motion. For mechanical Hamiltonian functions --the kinetic energy plus the potential energy on a semi-Riemannian configuration space $(Q,g)$--, we completely characterize polynomial bundle maps arising from symmetric $(k+1,0)$-tensor fields and derive the conditions ensuring their fouling nature. Several explicit examples on the Euclidean plane and on the 2-sphere illustrate the method.