  In 1993, P. Rosenau and J. M. Hyman introduced and studied
Korteweg-de-Vries-like equations with nonlinear dispersion admitting compacton
solutions, $u_t+D_x^3(u^n)+D_x(u^m)=0$, $m,n>1$, which are known as the
$K(m,n)$ equations. In the present paper we consider a slightly generalized
version of the $K(m,n)$ equations for $m=n$, namely,
$u_t=aD_x^3(u^m)+bD_x(u^m)$, where $m,a,b$ are arbitrary real numbers. We
describe all generalized symmetries and conservation laws thereof for $m\neq
-2,-1/2,0,1$; for these four exceptional values of $m$ the equation in question
is either completely integrable ($m=-2,-1/2$) or linear ($m=1$) or trivial
($m=0$). It turns out that for $m\neq -2,-1/2,0,1$ there are only three
symmetries corresponding to $x$- and $t$-translations and scaling of $t$ and
$u$, and four nontrivial conservation laws, one of which expresses the
conservation of energy, and the other three are associated with the Casimir
functionals of the Hamiltonian operator $\mathfrak{D}=aD_x^3+bD_x$ admitted by
our equation. Our result, \textit{inter alia}, provides a rigorous proof of the
fact that the K(2,2) equation has just four conservation laws found by P.
Rosenau and J. M. Hyman.
