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Well, your objective isn't an impossible one, but it'll take some doing. The "non-student" criteria is probably the easiest one to manage, actually, because although there is certainly a lot of student housing around the University and near downtown, you'll encounter few if any students renting more than a couple of miles from the campus.
Probably what is going to be most difficult is to find a rental property where you can have large dogs. It's funny; Madison is a pretty dog-friendly place in general, but only if you own your own place. If you want to rent with a dog, it's tough to find a spot. I'd recommend that you get in touch with the Dane County Humane Society for starters; they keep a list of property owners who allow dogs, and you'll probably encounter less frustration if you start by focusing there.
Unfortunately, it's also true that the majority of rental housing in Madison is going to be in apartments or townhouses or duplexes rather than in single-family units. One possibility might be to check with a real estate agent, as sometimes a family that has had to relocate and is having difficulty finding a buyer for their home may be willing to rent the house to cover the mortgage costs.
Good luck in any case. Madison is an extremely tenant-friendly community, and once you find a place to rent, you'll be in good shape. I hope you find what you're looking for!
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