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We theoretically study the profile of a supercurrent in two-dimensionalJosephson junctions with Rashba-Dresselhaus spin-orbit interaction (RDSOI) inthe presence of a Zeeman field. Through investigating self-biased supercurrent(so called $\varphi_0$-Josephson state), we obtain explicit expressions for thefunctionality of the $\varphi_0$ state with respect to RDSOI parameters($\alpha,\beta$) and in-plane Zeeman field components ($h_x,h_y$). Our findingsreveal that, when the chemical potential ($\mu$) is high enough compared to theenergy 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 Zeemanfield with unequal components, i.e., $|h_x|\neq |h_y|$, however, can counteractand nullify the destructive impact of equal-strength RDSOIs (for one typeonly), 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-planeZeeman field, i.e., $h_xh_y$, which is absent in the $\mu \gg \Delta$ limit.Furthermore, our results of critical supercurrents demonstrate that thepersistent spin helices can be revealed in a high enough chemical potentialregime $\mu\gg \Delta$, while an opposite regime, i.e., $\mu\sim\Delta$,introduces an adverse effect. In the ballistic regime, the "maximum" of thecritical supercurrent occurs at $|\alpha|=|\beta|$ and the Zeeman field canboost this feature. The presence of disorder and nonmagnetic impurities changethis picture drastically so the "minimum" of the critical supercurrent occursat and around the symmetry lines $|\alpha|=|\beta|$.

Critical supercurrent and $\phi_0$ state for probing a persistent spin helix