Differential birational equivalence

Suppose the base field algebraically closed and of zero characteristic.

There are two fascinating questions in the intersection of ring theory and algebraic geometry (for which an excellent discussion is Y. Berest, G. Wilson, "Differential isomorphism and equivalence of algebraic varieties" MR2079372, whose terminology I use here).

• (Differential isomorphism) Let $X$ and $Y$ be two irreducible affine varieties. If they have isomorphic rings of differential operators $mathcal{D}(X) simeq mathcal{D}(Y)$, are they isomorphic? In general, what this says about the relation between $X$ and $Y$?

• (Differential equivalence) Let $X$ and $Y$ be two irreducible affine varieties. If $mathcal{D}(X)$ and $mathcal{D}(Y)$ are Morita equivalent, what this says about the relation between $X$ and $Y$?

It can be shown without much difficulty that for an affine irreducible $X$, $mathcal{D}(X)$ is an Ore domain, and hence admits a skew-field of fractions, $mathbb{D}(X)$.

• Question 1: If $X$ and $Y$ are two irreducible affine varieties and $mathbb{D}(X) simeq mathbb{D}(Y)$, are they isomorphic? In general, what this implies about the relation between $X$ and $Y$?

The only work I know of in this direction is J. P. Bell and A. Smoktunowicz, "Rings of differential operators on curves", MR3004084, for $X$ and $Y$ curves.

• Question 2: How we can distinguish $mathbb{D}(X)$ from $mathbb{D}(mathbb{A}^n)$; i.e., show that it is not a Weyl field (the skew-field of fractions of the Weyl algebra)?

I know of something about Question 2 related to the famous Gelfand-Kirillov Conjecture (cf. J. Alev, A. Ooms, M. Van den Bergh, "A class of counterexamples to the Gel’fand-Kirillov conjecture", MR1321564).