= Solution
The steady incompressible <shearing sheet> equation is
$$
\mathbf u\cdot\nabla\mathbf u+2\Omega\mathbf e_z\times\mathbf u=-\rho^{-1}\nabla p+2\Omega Sx\mathbf e_x.
$$
The final term follows from the local tidal potential $-\Omega Sx^2$. For the given linear velocity, the advective acceleration is $-\alpha\beta(x\mathbf e_x+y\mathbf e_y)$, while the <Coriolis acceleration> is $2\Omega\beta x\mathbf e_x+2\Omega\alpha y\mathbf e_y$. Equating components gives
$$
\frac{p_x}\rho=[\alpha\beta+2\Omega(S-\beta)]x,\qquad\frac{p_y}\rho=\alpha(\beta-2\Omega)y.
$$
The cross derivatives vanish, so the <pressure Hessian of an elliptical shearing-sheet vortex> integrates to
$$
\boxed{\frac p\rho=\frac12Ax^2+\frac12By^2+\text{constant},\quad A=\frac{S^2}{(r-1)^2}-\frac{2\Omega S}{r-1},\quad B=\frac{S^2}{(r-1)^2}-\frac{2\Omega S}{r(r-1)}.}
$$
The additive constant is fixed by exterior matching or the <pressure> reference. No zero-pressure free-boundary condition is imposed on an incompressible vortex patch: it is embedded in a surrounding fluid.
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