  By a semi-Lagrangian change of coordinates, the hydrostatic Euler equations
describing free-surface sheared flows is rewritten as a system of quasilinear
equations, where stability conditions can be determined by the analysis of its
hyperbolic structure. This new system can be written as a quasi linear system
in time and horizontal variables and involves no more vertical derivatives.
However, the coefficients in front of the horizontal derivatives include an
integral operator acting on the new vertical variable. The spectrum of these
operators is studied in detail, in particular it includes a continuous part.
Riemann invariants are then determined as conserved quantities along the
characteristic curves. Examples of solutions are provided, in particular
stationary solutions and solutions blowing-up in finite time. Eventually, we
propose an exact multi-layer $\mathbb{P}_0$-discretization, which could be used
to solve numerically this semi-Lagrangian system, and analyze the eigenvalues
of the corresponding discretized operator to investigate the hyperbolic nature
of the approximated system.
