In this work we prove the existence of an autonomous Hamiltonian vector field in W-1,W-r(T-d; R-d) with r < d - 1 and d >= 4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.