  A new two-component system with cubic nonlinearity and linear dispersion:
\begin{eqnarray*} \left\{\begin{array}{l}
m_t=bu_{x}+\frac{1}{2}[m(uv-u_xv_x)]_x-\frac{1}{2}m(uv_x-u_xv), \\
n_t=bv_{x}+\frac{1}{2}[ n(uv-u_xv_x)]_x+\frac{1}{2} n(uv_x-u_xv),
\\m=u-u_{xx},~~ n=v-v_{xx}, \end{array}\right. \end{eqnarray*} where $b$ is an
arbitrary real constant, is proposed in this paper. This system is shown
integrable with its Lax pair, bi-Hamiltonian structure, and infinitely many
conservation laws. Geometrically, this system describes a nontrivial
one-parameter family of pseudo-spherical surfaces. In the case $b=0$, the
peaked soliton (peakon) and multi-peakon solutions to this two-component system
are derived. In particular, the two-peakon dynamical system is explicitly
solved and their interactions are investigated in details. Moreover, a new
integrable cubic nonlinear equation with linear dispersion \begin{eqnarray*}
m_t=bu_{x}+\frac{1}{2}[m(|u|^2-|u_x|^2)]_x-\frac{1}{2}m(uu^\ast_x-u_xu^\ast),
\quad m=u-u_{xx}, \end{eqnarray*} is obtained by imposing the complex conjugate
reduction $v=u^\ast$ to the two-component system. The complex valued $N$-peakon
solution and kink wave solution to this complex equation are also derived.
