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Sprinkle rabbit (or chicken) on both sides with 1/4 teaspoon each salt and pepper. Heat 2 tablespoons oil in a large, wide pot over medium heat. Cook the meat, in batches if necessary, turning 2 to 3 times, until golden brown but not cooked through, 4 to 8 minutes total. Remove to a shallow bowl. Pour wine into the pan, increase heat to medium-high and cook until the wine is reduced to a couple of tablespoons, about 2 minutes. Add the reduction to the bowl with the meat; cover to keep warm.
Heat the remaining 2 tablespoons oil in the pot over medium heat. Add onion and cook, stirring gently, until soft but not brown, 2 to 3 minutes. Add potatoes, celery and carrots and cook, stirring often, until the vegetables start to soften, 5 to 6 minutes. Add tomatoes, olives, capers and pine nuts and cook, stirring, until the tomatoes start to break down, about 5 minutes.
Return the meat and all the juice to the pot, nestling the pieces into the vegetables. (They should just fit and the vegetable sauce should lap up around the edges.) Cover and continue cooking gently over medium-low heat for 35 minutes (or just 20 minutes if using chicken).
Meanwhile, whisk vinegar and honey in a small saucepan; bring to a boil over high heat. Reduce heat to medium-low and simmer until reduced by half, 5 to 8 minutes. When the rabbit has cooked for about 35 minutes (or 20 minutes for chicken), stir the vinegar reduction into the pot. Increase heat to medium and cook, uncovered, for 10 minutes more. Stir in the remaining 1/4 teaspoon salt.
Serve the meat with the vegetable sauce.