= Solution
For steady flow the given equation gives $\nabla H=-\rho\,\omega\times u$. Dotting with $u$ yields $u\cdot\nabla H=0$, so \b[$H$ is constant along each <streamline>]. It can have different values on different streamlines. Dotting with $\omega$ also makes it constant along vortex lines where they are defined.
If the flow is additionally irrotational, $\omega=0$ and hence $\nabla H=0$. Thus \b[$H$ is one constant on each <connected> fluid region], the stronger global Bernoulli conclusion.
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