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The purpose of this study is to systematically identify all feasible trajectories following a gravity-assisted flyby that immediately return to the flyby body with no intermediate maneuvers. Every class of possible transfer angles is considered including even-nπ, odd-nπ, and generic return orbits. Lambert's problem is solved for a desired timeof-flight range allowing the possibility for multiple spacecraft and celestial body revolutions. The solutions are expressed geometrically, and the resulting velocity diagram is a mission-planning tool with potential applications that include cycler trajectories and planetary moon tours. The generalized free-return solutions can be used to construct loitering orbits about one celestial body or transfers between multiple bodies. Several previously documented cycler trajectories are improved using the discussed solutions. Nomenclature a = semimajor axis E = eccentric anomaly F = focus of a transfer ellipse M = number of complete revolutions made by the celestial body N = number of complete revolutions made by the spacecraft r = position vector T = period t = time v = velocity vector α, β, S = intermediate variables for Lambert's equation γ = flight-path angle θ, δ, r 1 , r 2 , c = input geometry for Lambert's equation; see Fig. 2a µ = gravitational parameter of the primary ψ = referencing angle for generic returns; see Fig. 12 Subscripts B = celestial body F = full-revolution return f = final H = half-revolution return r = radial component v = vacant focus θ = transverse component 0 = initial ∞ = relative to celestial body before or after flyby

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