  We establish blow-up results for systems of NLS equations with quadratic
interaction in anisotropic spaces. We precisely show finite time blow-up or
grow-up for cylindrical symmetric solutions. With our construction, we moreover
prove some polynomial lower bounds on the kinetic energy of global solutions in
the mass-critical case, which in turn implies grow-up along any diverging time
sequence. Our analysis extends to general NLS systems with quadratic
interactions, and it also provides improvements of known results in the radial
case.
