We consider the multi-dimensional generalisation of the problem of a sphere, with axi-symmetric mass distribution, that rolls without slipping or spinning over a plane. Using recent results from García-Naranjo \cite{LGN18} and García-Naranjo and Marrero \cite{GNMarr2018}, we show that the reduced equations of motion possess an invariant measure and may be represented in Hamiltonian form by Chaplygin's reducing multiplier method. We also prove a general result on the existence of first integrals for certain Hamiltonisable Chaplygin systems with internal symmetries that is used to determine conserved quantities of the problem.