  We study the asymptotic dynamics for solutions to a system of nonlinear
Schr\"odinger equations with cubic interactions, arising in nonlinear optics.
We provide sharp threshold criteria leading to global well-posedness and
scattering of solutions, as well as formation of singularities in finite time
for (anisotropic) symmetric initial data. The free asymptotic results are
proved by means of Morawetz and interaction Morawetz estimates. The blow-up
results are shown by combining variational analysis and an ODE argument, which
overcomes the unavailability of the convexity argument based on virial-type
identities.
