By an eikonal Hamilton-Jacobi equation I mean a PDE of the form $$H(x, du(x)) = 0$$ where the Hamiltonian $H: T'M \to \mathbb R$ is given and we are solving for a function $u: M \to \mathbb R$. (Actually, in the application I'm interested in, I am more concerned with when $u$ is a $1$-form and so $H$ takes in a $2$-form, but already the case that $u$ is scalar seems nontrivial enough.)
The prototypical example is of course the eikonal equation $$|du| = 1$$ on $\Omega \subseteq \mathbb R^d$. The local Cauchy problem for the eikonal equation is well-posed and has a straightforward solution: for an embedded oriented hypersurface $N \subset \mathbb R^d$, we can find a solution $u$ on a tubular neighborhood of $N$, by letting $u(x)$ be the signed distance from $x$ to $N$. Conversely, any solution of the eikonal equation must arise in this way. It follows from this, and the fact that the only foliations of $\mathbb R^d$ into lines are rigid motions of the usual foliation $\mathbb R^d = \mathbb R^{d - 1} \times \mathbb R$, that the only (sufficiently regular) solutions of the eikonal equation on the entire space $\mathbb R^d$ are affine functions. In particular, the space of global solutions of the eikonal equation is $d + 1$-dimensional.
However, this seems particular to euclidean space. If instead we considered the eikonal equation on hyperbolic space $\mathbb H^d$, then there are lots of geodesic foliations $\mathscr F$ and they all give a global solution of the eikonal equation, namely $u$ such that $du$ is a calibration of $\mathscr F$. But of course we have a diffeomorphism $\mathbb H^d \to \mathbb R^d$, which sends the eikonal equation to an eikonal Hamilton-Jacobi equation.
So this leads me to ask: What is known about the space $S$ of solutions of an eikonal Hamilton-Jacobi equation on the entire space $\mathbb R^d$? Under what hypotheses is $S$ finite-dimensional (and under those hypotheses, can we bound its dimension explicitly)? If the general case is too hard, what can we say if we restrict to the equation $$|du|_g = 1$$ where $g$ is a Riemannian metric on $\mathbb R^d$?
