Compacton equations and integrability: the Rosenau-Hyman and Cooper-Shepard-Sodano equations

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, $$D_t(u)+u^{l-2}{D}_x(u)+\alpha p{D}_x(u^{p-1}{u}_x^2)+2\alpha D_x^2(u^p{u}_x)=0\mathrm{.}$$ 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 $CSS$.