  We exhibit a new decomposition of the nonlinearity for the Muskat equation
and use it to commute Fourier multipliers with the equation. This allows to
study solutions with critical regularity. As a corollary, we obtain the first
well-posedness result for arbitrary large data in the critical space
$\dot{H}^2(\mathbb{R}^2)\cap W^{1,\infty}(\mathbb{R}^2)$. Moreover, we prove
the existence of solutions for initial data which are not Lipschitz.
