Reciprocal link for a coupled Camassa–Holm type equation

• We construct a reciprocal transformation for a coupled Camassa–Holm type equation proposed by Geng and Xue.
• The transformed coupled Camassa–Holm type system is a reduction of the first negative flow in a modified Drinfeld–Sokolov III hierarchy.
• The Lax pair and bi-Hamiltonian structure behaviors of the coupled Camassa–Holm type equation under the reciprocal transformation are analyzed.

A coupled Camassa–Holm type equation is linked to the first negative flow in a modified Drinfeld–Sokolov III hierarchy by a transformation of reciprocal type. Meanwhile the Lax pair and bi-Hamiltonian structure behaviors of this coupled Camassa–Holm type equation under the reciprocal transformation are analyzed.