Car traffic, while useful to its users owing to high service quality at moderate cost, exerts negative impacts on the environment local and global, henceforth on human health. Ride-sharing services enable their participants as Users (riders) or Agents (drivers) to make better use of seats in cars and to save money costs as well as environmental impacts, at the expense of some additional times of dwelling for rider boarding and alighting, of waiting and of transaction with the service platform. Line ride-sharing (LRS) is a specific kind of service fostering service participation as User (rider) or Agent (driver), by aligning and channeling both car runs and rides, thereby increasing run frequency (factor phi) and ride load per car (factor omega). This article investigates the structural properties of an LRS service on a roadway link. From the stochastic properties of two flows of arrivals, Agents and Users, stem two physical laws about the number of users per agent run (a geometric distribution) and the wait time distributions according to the service Waiting Policy (WP) either Agent Waits (AW) or User Waits (UW). Then, service conditions ( omega, phi) determine the expected values of agent rewards and of wait times by role U and A, contributing to their utility functions as modal options faced to serviceneutral option N of non-participation. From the discrete choice of mode U, A or N by link users stem two economic laws: the inner-service split condition between U and A yields a relationship between frequency phi and cab load omega, while the outer-service split condition between N and service participation S={A,U} relates the service participation rate to omega. These laws determine traffic equilibrium between the three modes at the link level under exogenous price schedule (user fare and agent reward). Further on, from service management aimed to just operator's profit or to system optimum (first or second best) also considering demand surplus and environmental impacts, the optimal service prices are endogenized, yielding specific forms of the two economic laws. By combining them in a "demanded volume" function of cab load, the optimal states are straightforwardly obtained for the full range of link flows, according to waiting policy and regulation regime. A numerical experiment is performed to assess and compare the diverse service specifications with respect to the link flow.