  We consider the incompressible Euler equations on ${\mathbb R}^d$, where $d
\in \{ 2,3 \}$. We prove that:
  (a) In Lagrangian coordinates the equations are locally well-posed in spaces
with fixed real-analyticity radius (more generally, a fixed Gevrey-class
radius).
  (b) In Lagrangian coordinates the equations are well-posed in highly
anisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolev
regularity in the labels $a_2,...,a_d$.
  (c) In Eulerian coordinates both results (a) and (b) above are false.
