This paper characterizes all utility derived demand systems having Engel curves that are linear in both income and an arbitrary function of income. This class encompasses virtually all utility derived demand systems that have been estimated in the past using aggregate data with explicit treatment of the problem of aggregation across individuals. It includes extensions of the PIGLOG and PIGL classes that have similar properties to these classes, but allow for more general Engel curve shapes. This paper extends Gorman's study of these forms primarily by characterizing systems of rank two. The application of the characterized systems to problems of nesting, separability, flexibility, Engel curve analysis, estimation, and aggregation are briefly discussed.