  This paper gives an integrable hierarchy of nonlinear evolution equations. In
this hierarchy there are the following representative equations: \beqq & &
u_t=\pa^5_x u^{-{2/3}}, & & u_t=\pa^5_x\frac{(u^{-{1/3}})_{xx}
  -2(u^{-{1/6}})_{x}^2}{u}; & & u_{xxt}+3u_{xx}u_x+u_{xxx}u=0. \eeqq The first
two are in the positive order hierarchy while the 3rd one is in the negative
order hierarchy. The whole hierarchy is shown integrable through solving a key
$3\times 3 $ matrix equation. The $3\times3$ Lax pairs and their adjoint
representations are nonlinearized to be two Liouville-integrable canonical
Hamiltonian systems. Based on the integrability of $6N$-dimensional systems we
give the parametric solution of the positive hierarchy. In particular, we
obtain the parametric solution of the equation $u_t=\pa^5_x u^{-{2/3}}$.
Moreover, we give the traveling wave solution (TWS) of the above three
equations. The TWSs of the first two equations have singularity and look like
cusp (cusp-like), but the TWS of the 3rd one is continuous. For the 5th-order
equation, its parametric solution can not include its singular TWS. We also
analyse the Gaussian initial solutions for the equations $u_t=\pa^5_x
u^{-{2/3}}$, and $u_{xxt}+3u_{xx}u_x+u_{xxx}u=0.$ One is stable, the other not.
Finally, we extend the equation $u_t=\pa^5_x u^{-{2/3}}$ to a large class of
equations $ u_t=\partial_x^l u^{-m/n}, l\ge1, n\not=0, m,n \in \Z, $ which
still have the singular cusp-like traveling wave solutions.
