"Most of the time, elimination outside of the box is a symptom of a physical problem. You should have you cat undergo a full physical. The UTI may be cured, but also check for FUS,"
Absolutely. I've owned cats for more than 3 decades, and the first step for "inappropriate urination" is definitely a complete physical. Rule out or define any underlying problems, that sort of thing.
Beyond that, highly qualified behaviorists can sometimes work wonders. Sounds like your cat has been a really good pet. I vote you give him every opportunity to remain with you.
Hope things work out. Please let us know.
Please don't give up on your kitty yet.
Cats who have UTI's will often start to pee somewhere else, because they associate the pain of the infection while urinating to the litterbox. They are literally looking for somewhere less painful to pee.
The most important step, as others have already said, is a full medical exam to make sure that the medical problem is resolved and the pain goes away. If there was a problem with crystals in the urinary tract, your vet should have talked to you about dietary changes to avoid future similar problems.
Even then, your cat may still fear pain associated with the old litterbox, so it is a good idea to get a brand new box and put it in a new location.
If you have two cats, make sure you have two separate litterboxes. The rule is one box per cat to avoid accidents or territory issues.
Make sure you get a good enzymatic cleaner and thoroughly treat all areas where the cat has had accidents, in order to deter the cat from returning to that spot.
Finally, cats really don't pee out of spite. Spite is a human emotion, and cats aren't even cognitively capable of it, even though people like to attribute it to them.
If all else fails, consult a cat behavior specialist, as someone earlier in this discussion suggested. Usually behaviorists will charge for a single session in which they evaluate the problem and give you a very detailed plan to solve it. Often follow-up phone calls are included in the deal.
Good luck. I hope your kitty is feeling better, and that you find a solution soon. Cat pee in the house is no fun.