We provide a full characterization of a two-type optimal nonlinear income tax model where the single-crossing condition is violated due to an assumption that agents differ both in terms of market abilities and in terms of their needs for a work-related good. We set up a Pareto-efficient tax problem and analyze the entire second-best Pareto-frontier, highlighting several non-standard results, such as the possibility of income re-ranking relative to the laissez-faire and gaps in the Pareto-frontier.