  We study integrability --in the sense of admitting recursion operators-- of
two nonlinear equations which are known to possess compacton solutions: the
$K(m,n)$ equation introduced by Rosenau and Hyman
  \[ D_t(u) + D_x(u^m) + D_x^3(u^n) = 0 \; , \] and the $CSS$ equation
introduced by Coooper, Shepard, and Sodano,
  \[ D_t(u) + u^{l-2}D_x(u) + \alpha p D_x (u^{p-1} u_x^2) + 2\alpha D_x^2(u^p
u_x) = 0 \; . \] We obtain a full classification of {\em integrable $K(m,n)$
and $CSS$ equations}; we present their recursion operators, and we prove that
all of them are related (via nonlocal transformations) to the Korteweg-de Vries
equation. As an application, we construct isochronous hierarchies of equations
associated to the integrable cases of $CSS$.
