  We study the nonlinear Schr\"odinger system
  \[ \begin{cases}
  \displaystyle iu_t+\Delta
u-u+(\frac{1}{9}|u|^2+2|w|^2)u+\frac{1}{3}\overline{u}^2w=0,\\
  i\displaystyle \sigma w_t+\Delta w-\mu w+(9|w|^2+2|u|^2)w+\frac{1}{9}u^3=0,
\end{cases} \] for $(x,t)\in \mathbb{R}^n\times\mathbb{R}$, $1\leq n\leq 3$ and
$\sigma,\mu>0$. This system models the interaction between an optical beam and
its third harmonic in a material with Kerr-type nonlinear response. We prove
the existence of ground state solutions, analyse its stability, and establish
local and global well-posedness results as well as several criteria for
blow-up.
