The invention relates to a mechanism for converting a rotary motion originating from a rotary energy generator (1) into a longitudinal motion of a carriage (25) that is adapted for attachment to a covering, such that said rotary motion will result in a longitudinal displacement of the covering, the mechanism comprising one or more spindle sections (2, 3, 4, 5) drivingly connected at a longitudinal end of at least one of said spindle sections via a gear means (7; 9; 22, 23, 24, 27) to said rotary energy generator (1). The invention further relates to a drive system for coverings comprising mechanisms according to the invention. The invention further relates to a covering system for covering a surface portion such as a window, glass door or the like, of a building.