  We theoretically investigate the nonlinear response current of a
two-dimensional system under an in-plane magnetic field. Based on the extended
semiclassical theory, we develop a unified theory including both longitudinal
and transverse currents and classify contributions according to their scaling
with the relaxation time. Besides time-reversal-even contributions, we reveal a
previously unknown time-reversal-odd contribution to the Hall current, which
occurs in magnetic systems, exhibits band geometric origin, and is linear in
relaxation time. We show that the different contributions exhibit different
symmetry characters, especially in their angular dependence on the field
orientation, which can be used to distinguish them in experiment. The theory is
explicitly demonstrated in the study of the Rashba model. Our work presents a
deepened understanding of nonlinear planar transport, proposes approaches to
distinguish different contributions, and sheds light on possible routes to
enhance the effect in practice.
