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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 spaceswith fixed real-analyticity radius (more generally, a fixed Gevrey-classradius).  (b) In Lagrangian coordinates the equations are well-posed in highlyanisotropic spaces, e.g.~Gevrey-class regularity in the label $a_1$ and Sobolevregularity in the labels $a_2,...,a_d$.  (c) In Eulerian coordinates both results (a) and (b) above are false.

Contrast between Lagrangian and Eulerian analytic regularity properties of Euler equations