e intensively studied during the last fifty years [@LITTLE; @DOU; @VEKHT]. Recent advances in nanotechnology have made it possible for studying experimentally superconducting properties of thin films with different arrays of antidots, see for example, [@Motta1] and references therein. In particular, for the observation of the aforementioned effects, cylinders or antidots of small diameter are required. At the same time, the study of hollow thin-walled cylinders with macroscopic sizes in magnetic fields parallel to its axis has been much less well studied. It was expected that quantum phenomena cannot be observed in such samples because of the fact that one flux quanta for cylinders with a cross section area of $\approx 1$ cm$^2$ corresponds to a magnetic field about $10^{-7}$ Oe. In this case magnetization will be a smooth function of the magnetic field. However, experimental results obtained recently for thin-walled macroscopic cylinders do not agree with this expectation. Namely, in such Nb cylinders we succeeded in monitoring the magnetic moment of the current circulating in the walls and observed dc magnetic moment jumps even in fields much lower than $H_{c1}$ of the film itself [@Katz1]. So far it is not clear what mechanism is responsible for such flux jumps. Under an axial magnetic field the cylinder walls screen weak external fields, provided that $L\equiv Rd/\lambda^{2} \gg 1$, where $R$ is the cylinder radius, $d$ is the wall thickness, and $\lambda$ is the London penetration depth  [@DOU; @PG; @KITTEL]. Therefore, it is expected, that a dc magnetic field, $H_0$, will penetrate into the cylinder as soon as the current in the wall exceeds the critical current and no field penetration should be observed at lower fields. Only above $H_{c1}$, vortices created at the outer cylinder surface can move into the cylinder. For a magnetic field oriented perpendicular to the Nb film surface such vortex motion leads to flux jumps [@NOWAK; @STAM]. These flux jumps were interpreted as a thermomagnetic instability 