This study builds on Huntley and Ryan (2018) and related prior works on channels and estuaries by considering solutions for the same subtidal dynamics but with alternative integral flow constraints. Prior solutions do not have a truly two-dimensional (2D) flow field, as axial changes in the axial flow are implied. Three constraint types are considered: the Constant case with spatially constant density gradients constrained with section-integrated flow (as in prior works), Semi-Variable case with a constant axial density gradient and laterally variable lateral density gradient constrained with depth-integrated lateral flow, and Variable case with spatially variable density gradients in both directions constrained with depth-integrated axial and lateral flows. The Semi-Variable and Variable cases can produce solutions with truly 2D flow if the depth-integrated lateral flow is set to zero everywhere. Differences among solutions are illustrated with idealized and realistic applications from Huntley and Ryan (2018). For the idealized application, the Constant case and the Semi-Variable case with 2D flow have clear differences in axial velocity and stark contrasts in lateral flow (and density gradients). For the Nares Strait application, the Variable case with observed depth-averaged axial and lateral velocities is best able to represent the fastest observed down-channel velocities, the weak reversed flow on one side of the channel, and the observed lateral flow structure. Overall, selecting different integral constraints on flow conspicuously changes the subtidal flow, opens up new possibilities for truly 2D flow solutions, and provides additional flexibility for representing observed conditions in realistic situations.