Squared eigenfunction symmetry of the DΔmKP hierarchy and its constraint

In this paper, squared eigenfunction symmetry of the differential‐difference modified Kadomtsev–Petviashvili (DΔmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DΔmKP hierarchy, together with their adjoint forms, give rise to the positive relativistic Toda (R‐Toda) hierarchy. An invertible transformation is given to connect the positive and negative R‐Toda hierarchies. The positive R‐Toda hierarchy is reduced to the differential‐difference Burgers hierarchy. We also consider another DΔmKP hierarchy and show that its squared eigenfunction symmetry constraint gives rise to the Volterra hierarchy. In addition, we revisit the Ragnisco–Tu hierarchy which is a squared eigenfunction symmetry constraint of the differential‐difference Kadomtsev–Petviashvili (DΔKP) system. It was thought the Ragnisco–Tu hierarchy did not exist one‐field reduction, but here we find a one‐field reduction to reduce the hierarchy to the Volterra hierarchy. Besides, the differential‐difference Burgers hierarchy is also investigated in the Appendix. A multidimensionally consistent three‐point discrete Burgers equation is given.