Looks to me like you have an ABS control relay where the wiper relay should be. You really need to list what E39 that you own, year, model, engine, auto or manual trans etc.
In some cases I have read that giving the wiper motor a tap with a hammer gets it going, it’s worth a try but you need to remove some bits to get to it, that’s the drawback. Sometimes it’s the wiper linkage that’s binding, you gotta look at everything. A coworkers 2008 X5 wipers were not working a few years back, I found that the linkage was loose, adjusted it, tightened everything, and they still work to this day.
Hopefully that was all it was. You should look through old posts of e39 wiper issues just to get an idea of all the different things that can cause non-operation situations.
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