Squared eigenfunction symmetry of the DΔmKP hierarchy and its constraint

In this paper, squared eigenfunction symmetry of the differential-difference modified Kadomtsev-Petviashvili (D Delta mKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the D Delta mKP 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 D Delta mKP 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 (D Delta KP) 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.