  We study certain two-dimensional variational systems, namely pluri-Lagrangian
systems on the root lattice $Q(A_{N})$. Here, we follow the scheme which was
already used to define two-dimensional pluri-Lagrangian systems on the lattice
$\mathbb{Z}^{N}$ and three-dimensional pluri-Lagrangian systems on the lattice
$\mathbb{Z}^{N}$ as well as on $Q(A_{N})$. We will show that the
two-dimensional pluri-Lagragian systems on $Q(A_{N})$ are more general than the
ones on $\mathbb{Z}^{N}$, in the sense that they can encode several different
pluri-Lagrangian systems on $\mathbb{Z}^{N}$. This also means that the
variational formulation of several systems of certain hyperbolic equations,
so-called quad-equations, can be obtained from one and the same
pluri-Lagrangian system on $Q(A_{N})$.
