= Solution
For incompressible inviscid flow without body force,
$$
\nabla\cdot u=0,\qquad \partial_tu+(u\cdot\nabla)u=-\rho^{-1}\nabla p.
$$
Taking curl gives
$$
\partial_t\omega+(u\cdot\nabla)\omega=(\omega\cdot\nabla)u,\qquad \omega=\nabla\times u,
$$
equivalently $\partial_t\omega=\nabla\times(u\times\omega)$.
Direct <differentiation> of the stated field gives $\omega=u$. Hence $u\times\omega=0$, so the <vorticity equation> gives $\partial_t\omega=0$ and the field is steady. Finally
$$
(u\cdot\nabla)u=\nabla(|u|^2/2)-u\times\omega=\nabla(|u|^2/2).
$$
Euler's equation therefore implies
$$
\nabla\left(p+\frac12\rho|u|^2\right)=0,
$$
so $H$ is spatially uniform.
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