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 Dt(u) +Dx(um) + D3x(un) = 0, and the CS Sequation introduced by Coooper, Shepard, and Sodano, Dt(u) + ul−2 Dx(u) + αpDx(up−1u2x) + 2αD2x(upux) = 0. We obtain a full classification of 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. [ABSTRACT FROM AUTHOR]