  In this paper, we study the solutions to the energy-critical quadratic
nonlinear Schr\"odinger system in ${\dot H}^1\times{\dot H}^1$, where the sign
of its potential energy can not be determined directly. If the initial data
${\rm u}_0$ is radial or non-radial but satisfies the mass-resonance condition,
and its energy is below that of the ground state, using the
compactness/rigidity method, we give a complete classification of scattering
versus blowing-up dichotomies depending on whether the kinetic energy of ${\rm
u}_0$ is below or above that of the ground state.
