I take it that you have 4 12v batteries in parallel.
If so you will probably not be able to use the desulphator if designed for 24v, it all depends on the internal circuit.
Most of these things produce a string of fast high voltage pulses and it may well still work at 12v but you can't be sure that it will pulse at the lower voltage.
The solar panel is probably to supply the loss caused by the pulser and will still put a few mA into a 12v battery.
Alternatitively it may be a scam and is only a trickle charger, in which case it may work as well at 12v.
Don't think I was correcting your spelling, I am in UK and we spell a lot of things differently, doesn't mean that you were wrong.
Flux