In the second part of the talk, we concentrate on Lie reductions of the multi-layer quasi-geostrophic problem and its exact invariant solutions. After classifying one- and two-dimensional subalgebras of the Lie invariance algebra, we exhaustively study codimension-one and -two Lie reductions. Notably, among invariant submodels of the original nonlinear model, we obtain uncoupled systems of well-known linear equations, including the Helmholtz, modified Helmholtz, Laplace, Klein-Gordon, Whittaker, Bessel and linearized Benjamin-Bona-Mahony equations. Integration of these systems significantly depends on spectral properties of the model's vertical coupling matrix, which we also revisit in detail. As a result, we construct wide families of exact solutions, including rediscovered representations of stationary and travelling baroclinic Rossby waves, coherent baroclinic eddies, hetons and localized dipolar vortices. We illustrate the physical relevance of obtained solutions using real-world geophysical data for a three-layer ocean model.

This talk is based on the joint work with A. Bihlo and R. Popovych: 
Symmetry analysis and exact solutions of multi-layer quasi-geostrophic problem, J. Nonlinear Sci. 36 (2026), 113, 82 pp, arXiv:2603.26959.