In this paper, squared eigenfunction symmetry of the differential-difference modified Kadomtsev–Petviashvili (DmKP) hierarchy and its constraint are considered. Under the constraint, the Lax triplets of the DmKP 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 DmKP 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 (DKP) 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.

中文翻译：

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DmKP 层次结构的平方特征函数对称性及其约束

在本文中，考虑了微分差分修正Kadomtsev - Petviashvili (D mKP) 层次结构的平方特征函数对称性及其约束。在约束下，D mKP 层次结构的 Lax 三元组连同它们的伴随形式，产生了正相对论 Toda (R-Toda) 层次结构。给出了一个可逆变换来连接正负 R-Toda 层次结构。正 R-Toda 层次结构被简化为差分-差分 Burgers 层次结构。我们还考虑了另一个 D mKP 层次结构，并表明其平方特征函数对称约束产生了 Volterra 层次结构。此外，我们重新审视了 Ragnisco-Tu 层次结构，它是差分差分 Kadomtsev-Petviashvili (DKP) 系统。人们认为 Ragnisco-Tu 层次结构不存在单场归约，但在这里我们找到了一个单场归约以将层次结构归约到 Volterra 层次结构。此外，附录中还研究了差分-差分 Burgers 层次结构。给出了多维一致的三点离散 Burgers 方程。