Off-axis elliptic Gaussian beams with an intrinsic orbital angular momentum

We discuss paraxial light beams composed of decentered Gaussian beams, with their phase selected in a special way so that their superposition is invariant as it propagates in free space, retaining its cross-section shape. By solving a system of five nonlinear equations, a superposition is constructed that forms an invariant off-axis elliptic Gaussian beam. An expression is obtained for the orbital angular momentum of this beam. It is shown that it consists of two components. The first of them is equal to the moment relative to the center of the beam and increases with increasing ellipticity. The second one quadratically depends on the distance from the center of mass to the optical axis (an analogue of Steiner's theorem). It is shown that the orientation of the ellipse in the transverse plane does not affect the normalized orbital angular momentum.