  We theoretically study the profile of a supercurrent in two-dimensional
Josephson junctions with Rashba-Dresselhaus spin-orbit interaction (RDSOI) in
the presence of a Zeeman field. Through investigating self-biased supercurrent
(so called $\varphi_0$-Josephson state), we obtain explicit expressions for the
functionality of the $\varphi_0$ state with respect to RDSOI parameters
($\alpha,\beta$) and in-plane Zeeman field components ($h_x,h_y$). Our findings
reveal that, when the chemical potential ($\mu$) is high enough compared to the
energy gap ($\Delta$) in superconducting electrodes, i.e., $\mu \gg \Delta$,
RSOI and DSOI with equal strengths ($|\alpha|=|\beta|$) cause vanishing
$\varphi_0$ state independent of magnetization and the type of RDSOI. A Zeeman
field with unequal components, i.e., $|h_x|\neq |h_y|$, however, can counteract
and nullify the destructive impact of equal-strength RDSOIs (for one type
only), where $\mu\sim\Delta$, although $|h_x|= |h_y|$ can still eliminate the
$\varphi_0$ state. Remarkably, in the $\mu\sim\Delta$ limit, the $\varphi_0$
state is proportional to the multiplication of both components of an in-plane
Zeeman field, i.e., $h_xh_y$, which is absent in the $\mu \gg \Delta$ limit.
Furthermore, our results of critical supercurrents demonstrate that the
persistent spin helices can be revealed in a high enough chemical potential
regime $\mu\gg \Delta$, while an opposite regime, i.e., $\mu\sim\Delta$,
introduces an adverse effect. In the ballistic regime, the "maximum" of the
critical supercurrent occurs at $|\alpha|=|\beta|$ and the Zeeman field can
boost this feature. The presence of disorder and nonmagnetic impurities change
this picture drastically so the "minimum" of the critical supercurrent occurs
at and around the symmetry lines $|\alpha|=|\beta|$.
