A point on the planet's equator that is "stationary with respect to the sun" is merely stationary with respect to a line joining it to the sun. That line is a radius of the orbit, sweeping round the sun at orbital velocity.
copyright, credit: forum cosmos quest, off google images.
Posted these images just as an illustration because they show features not rotating, not concentric with the cg of the rings., possibly pointing nearly parallel to the Sun. Stationary effects generated and traces left, during the passage of Saturn on its orbit, with the zero velocity area at noon somehow generating different reflections?
Going to Jupiter here and the carousel, (you got to give leeway to an amateur), Considering the Big red Spot. Every 5 hours, there would be a difference, peak and valley, of absorption and reflection of energy. This would be a
standing wave like feature, since it generated by an object with zero velocity, but varying energy intensity.
This "wave"*, standing in the ecliptic should have a wavelength of ~ 140 000 m for Jupiter and ~120 000 or fraction, multiples thereof for Saturn respectively. Vary with the seasons of course to have the equator align in different ways with the ecliptic.
These would be kind of standing compression waves, with peak energy every 5 hours.
Just pointing out the similarities: the spokes spacings are commensurate with features that relate to the radius of the orbit, not the rotation, A feature that would be created by a recurring condition (like zero velocity), as the planet passes along.
* The lifetime of this wave would be 80 minutes, more if any resonances are generated. (spooky action at a distance).