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Abstract The compatible trios of two-component homogeneous Hamiltonian operators were classified and some bi-Hamiltonian systems were constructed by Lorenzoni et al (2018 J. Phys. A: Math. Theor. 51 045202). In this paper, we study three two-component bi-Hamiltonian systems proposed by them. By means of the prolongation structure technique, we construct the missing Lax representations for those systems and confirm their integrability. Furthermore, we explore the possible connections between those systems and the known integrable systems.
Zang et al. (Sat,) studied this question.