We characterise the conditions for the existence of a mixed system of finance and provision of health-care by a social planner and a private market coexists. Private insurance/provision and public health-care provision cannot be consumed together, each individual has a certain probability of being sick or healthy, they have different incomes and the richer they are, the more health-care services they are willing to buy. Everybody contributes through a general linear income taxation to the finance of the public provision of health-care. Implicitly income is not observable by the social planner. The selection of users is endogenous. The quantity of health-care service supplied by the public sector suffers from congestion. We derive the conditions for existence of a mixed system and we characterise the impact of an increase or decrease of resources directed to the public system.