One thing you can do for starters is turn off the mesh with mesh=false since that keyword argument is Boolean. You can also improve the smoothness of the surface by setting plot_points higher than it's default of 40 in each direction.
If the surface could always be solved for each independent variable, one way to draw contour-like lines would be setting one variable to a constant, solving for the second in terms of the third and using parametric_plot. Since you can't do that for this surface, here's a workaround using a series of implicit_plot3d for lines with constant y-value:
for i in [ .1*j - .6 for j in range(0,13)]:

Here's a link to a live example. The function threejs at the end is a SageMathCell alternative to show.