I truly hope this was a one-off complaint, that you can patch things up with your neighbors and can all live happily ever-after.
But just in case this was only an opening salvo in a neighborhood campaign, be prepared.
First, as Faltrad says, respond to the landlord - or whoever signed the letter - in writing, explaining that your dog was very ill, and that the behavior was unusual. If your dog needed vet attention for the diarrhea, include a statement from the vet. Include steps you will take to prevent a similar occurance.
Second, if you haven't already done so, join the Mieterverband. Clarify with them if a one-off incidend can lead to termination of your lease.
Third, as Odile mentions - make sure you know exactly what your dog is doing. Keep a diary when you are home, try recording when you are away. You can bet your neighbors are doing the same.
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When I received a complaint here is what I did:
I first got my trainer on my side. I have been training with her for many years, she knows my dogs well, their strengths and weaknesses. We had already been working on the Belltie's love of the sound of his own voice, so we had an action plan to hand. If you don't have a trainer, now would be the time to get a pro on your side - not that this one-off is a training issue, but it adds to your reputation as a responsible owner .And you'd have a Swiss on your side.
My trainer came to the house to do some 'in situ' sessions - we did these outside in the garden for all the world (read: neighbors) to see. I made it very clear that I was taking steps to address the problem.
The complaint I received was anonymous, via the Quartierverwaltung - so I have no idea who was behind it. (Have my suspicions though.) I wrote a letter to the Quartierverwaltung, apologizing for disturbing anyone and stating that we were in training to address the problem. I also outlined the immediate management step I was taking (the miscreant and I are in training, I will keep windows closed, etc.)
My trainer offered to follow through with any neighbor who wished to discuss the situation; in the letter I gave her contact details. Not surprising, no one had the courage to take her up on it.
The response letter to the Verwaltung got the neighbors off my back for the short term, but there is still the issue of dog-hating neighbors to deal with. This is a very difficult problem. If the dog two streets away barks, it's my fault. If the neighbor's cat leaves poo in the Spielplatz, it's my fault. If the cowbells in the farm behind us are too loud, it's my fault.
I own my home, so they can't throw me out (well, not without deportation proceedings...) but they have made life pretty miserable.
Although the Belltie is much better now, I still have to have proof at all times that there was no barking - as there is no way to defend against an accusation if there are no witnesses. I am pretty much under 'house arrest', I cannot leave the dogs alone at anytime - and since I do not have a sitter, that means I cannot leave the house. No dinners out, no running errands, no meeting friends for coffee - nada.
But it's what I need to do to protect my dogs.
If you are working, perhaps you need to look at a doggy day care, where your dog would go to the carer's home during the day. Or, the more expensive option, get someone in to be with the dog all day.
Before receiving the complaint, I went out of my way to be a good neighbor. I sent wine and chocolates around whenever I adopted an new addition asking forbearance while we were in training, checked regularly to see if we were disturbing anyone - the answer was always 'no, not at all'. I was always the first one to volunteer to help out around the neighborhood, went along with whatever scheme the neighbor hatched for Quartier improvements. I went out of my way to be a good dog owner, good Auslander, good neighbor.
And look where it got me. I think being so nice backfired; the neighbors decided I was a push-over, and decided to turn the screws. It seems that in these parts 'Nachbarstreit' is a sort of game. And increasingly complaints against pets are used as a weapon in these games - neighbors know that you love your pet, they see an Achilles heel. (Just in case you think I'm paranoid , there was an article in a local paper on this subject mentioned in the 'Kept In Twos' thread in the pets section.)
Added to the usual neighborhood relations issues, Switzerland is indeed becoming quite dog-intolerant. It is getting very hard to give a dog a decent life here, given the public mood.
If this continues with your neighbors, there is really only one solution - move. (Which is what I have been trying to do for two years now. )
I know I sound bitter and cynical - but I have good reason to do so.
I really hope that it works out for you.