= Steady Euler flow with vorticity proportional to its stream function
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For a smooth planar <incompressible flow>, use $u=\psi_y$, $v=-\psi_x$, so its scalar <vorticity> is $\omega=-\Delta\psi$. If $\omega=\lambda\psi$ for a constant $\lambda$, the flow solves the steady constant-density <Euler equations> with a suitable pressure.
Indeed the vector identity for the convective acceleration gives
$$
(\mathbf u\cdot\nabla)\mathbf u=\nabla(|\mathbf u|^2/2)+\omega\nabla\psi.
$$
Therefore the pressure can be chosen so that the <Bernoulli function> satisfies $p/\rho+|\mathbf u|^2/2=C-\lambda\psi^2/2$. Its gradient is $-\omega\nabla\psi$, exactly the steady momentum equation. It is constant along each <streamline>, since the stream function is constant there.
For example $\psi=\sin x\cos y$ satisfies $\omega=2\psi$. Its pressure is $p_0-\rho\sin^2x/2+\rho\sin^2y/2$. Thus a rotational flow can have an explicit pressure and a streamline-dependent Bernoulli constant.
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