$\mathrm{PGL}(3)$-invariant integrable systems from factorisation of linear differential and difference operators

TL;DR

This paper develops a unified framework for constructing continuous and discrete PGL(3)-invariant integrable systems using spectral problem factorisation, generalising classical invariants to rank-3, and explores their geometric and hierarchical structures.

Contribution

It introduces explicit forms of PGL(3)-invariants, derives new integrable Boussinesq systems, and connects these to geometric lifting and hierarchy generation.

Findings

Explicit PGL(3) invariants generalising Schwarzian and cross-ratio.

Dualities in spectral problems underpin discretisation and consistency.

A hierarchy-generating PDE system with Lagrangian structure is established.

Abstract

In this paper, we present a unified approach to constructing continuous and discrete $\mathrm{PGL}(3)$-invariant integrable systems, formulated in terms of the common dependent variables $z_1,z_2$, from linear spectral problems and their factorisation. Starting from third-order spectral problems, we first provide explicit forms of the differential and difference invariants, generalising the Schwarzian derivative and cross-ratio to the rank-$3$ setting. The factorisation induces dualities among linear spectral problems, underlying the exact discretisation and multi-dimensional consistency of the associated Boussinesq systems. Then, we derive both continuous and discrete $\mathrm{PGL}(3)$-invariant Boussinesq systems, representing natural rank-$3$ generalisations of the Schwarzian KdV and cross-ratio equations. A geometric lifting-decoupling mechanism is developed to explain the reduction of these systems to the $\mathrm{PGL}(2)$-invariant Boussinesq equations. Finally, we derive a ${\mathrm{PGL}}(3)$-invariant system of generating PDEs together with its Lagrangian structure, in which the lattice parameters serve as independent variables, providing the generating PDE system for the Boussinesq hierarchy.