Is it permissible to buy a new car in which the wife is a partner in lieu of a debt owed to her by her husband, and what are the legitimate parameters for this partnership, and how is her share calculated when the car is sold in the future if it is legally registered in the husband's name?

You must repay your wife's debt in full. She is not obligated to share in the car's ownership in lieu of her debt. If the wife agrees to share in the car's ownership in lieu of her debt, there is no harm in that. In this case, the car becomes the property of both of you, according to each of your shares in the price, and she will have her share of the sale price upon selling it. This is a voluntary ownership partnership, and one partner is not permitted to dispose of the other partner's share without their permission. If the car is officially registered in your name, the wife's right to her share should be documented with a written contract and witnesses.