Place the rabbit pieces in a bowl with the wine-water mixture and marinate for 1 hour. Then drain and dry thoroughly with paper towels.
In a pan large enough to hold all the ingredients, heat 3 tablespoons of the olive oil over medium-high heat. Add the rabbit pieces and brown on both sides, about 5 to 7 minutes per side. As they brown, remove the pieces and set aside. When the rabbit is done, lower the heat to medium-low and stir the onion slices into the pan. Cook until the onion is limp and golden, but don't let it brown. Add the wine to the pan, bring to a simmer, then add the rabbit pieces, nestling them among the onions, and cook, covered, for about 30 minutes.
Put the remaining 3 tablespoons of oil in a separate frying pan and over medium-low heat gently simmer the pepper slices until they are thoroughly wilted and starting to brown; about 15 minutes uncovered and 15 minutes covered. Stir the peppers and their juices into the pan with the rabbit, add salt and pepper, to taste, and cook together for an additional 30 minutes.
Check the amount of liquid in the pan after 15 minutes, there should be a small amount of syrupy oil in the bottom of the pan, enough to keep the meat and vegetables from burning. Add a little more white wine, water, or broth if necessary to prevent sticking. Serve immediately.