Recently, by combining the tri-duality of Olver and Rosenau with the classification of second and third order homogeneous Hamiltonian operators, Lorenzoni and his collaborators classified compatible trios of two-component homogeneous Hamiltonian operators. Then, those trios were used to construct bi-Hamiltonian systems which, as they claimed, includes some known systems and some apparently new ones. In this talk, we consider those apparently new systems and construct their Lax representations. Furthermore, we explore the possible connections between these systems and the existing ones.