  The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of
two interconnected potential vorticities, each is governed by an active scalar
equation. These vorticities are linked through a distinctive combination of
their respective stream functions, which can be loosely characterized as a
parameterized blend of both Euler and shallow-water stream functions.
  In this article, we study (QS2L) in two directions: First, we prove the
existence and uniqueness of global weak solutions in the class of Yudovich,
that is when the initial vorticities are only bounded and Lebesgue-integrable.
The uniqueness is obtained as a consequence of a stability analysis of the
flow-maps associated with the two vorticities. This approach replaces the
relative energy method and allows us to surmount the absence of a velocity
formulation for (QS2L). Second, we show how to construct $m$-fold time-periodic
solutions bifurcating from two arbitrary distinct initial discs rotating with
the same angular velocity. This is achieved provided that the number of
symmetry $m$ is large enough, or for any symmetry $m\in \mathbb{N}^*$ as long
as one of the initial radii of the discs does not belong to some set that
contains, at most, a finite number of elements. Due to its multi-layer
structure, it is essential to emphasize that the bifurcation diagram exhibits a
two-dimensional pattern. Upon analysis, it reveals some similarities with the
scheme accomplished for the doubly connected V-states of the Euler and
shallow-water equations. However, the coupling between the equations gives rise
to several difficulties in various stages of the proof when applying
Crandall-Rabinowitz's Theorem. To address this challenge, we conduct a careful
analysis of the coupling between the kernels associated with the Euler and
shallow-water equations.
