We investigate the transformation of structured Laguerre–Gaussian (sLG) beams after passing through a cylindrical lens. The resulting beam, ab astigmatic structured Laguerre–Gaussian (asLG) beam, depends on quantum numbers (n,ℓ) and three parameters. Two of them are control parameters of the initial sLG beam, the amplitude ϵ and phase θ. The third one is the ratio of the Rayleigh length z0 and the focal length f of the cylindrical lens. It was theoretically revealed and experimentally confirmed that the asLG beam keeps the intensity shape of the initial sLG beam when the parameters satisfy simple conditions: ϵ is unity and the tangent of the phase parameter θ/2 is equal to the above ratio. We also found sharp bursts and dips of the orbital angular momentum (OAM) in the asLG beams in the vicinity of the point where the OAM turns to zero. The heights and depths of these bursts and dips significantly exceed the OAM maximum and minimum values of the initial sLG beam and are controlled by the radial number n.