  We study ordinary differential equations in the complex domain, and
investigate the absence of movable critical points (the Painlev\'e property)
for systems of non-autonomous polynomial quadratic differential equations in
two variables. We classify the systems having this property by studying, when
it is possible, some obstructions described by Malmquist, and by reducing them
to second-order equations of the first degree in one variable, for which the
analysis of Painlev\'e and Gambier can be applied. We find that, in many cases,
overcoming Malmquist's obstructions guarantees the absence of movable critical
points.
