8 0 3 6 9 12 15 0 2 4 6 8 10 12 0 3 6 9 12 15 0.0 0.4 0.8 1.2 H$_{0}$ = 51 A/m  = 0.644,  $_{c}$ = 36 0 T = 10 mK.  a) NbAu ( h ) h = H / H$_{0}$ h$_{max}$ h$_{max}$ b) NbAu ( h ) / $^{Ω}$ ( h max  ) h = H / H$_{0}$ 0 10 20 30 40 50 60 -15 -12 -9 -6 -3 0 0 3 6 9 12 -1.5 -1.0 -0.5 0.0 H$_{2}$ exp H$_{1}$ exp b) T = 7 mK d = 3.2  m R = 8.3  m $_{c}$ = 36 0 NbAu 4 M$^{dc}$ ( G ) H ( Oe )$^{H$_{2}$}$$^{calc}$ H$_{1}$ calc  = 0.644 $_{c}$ = 36 0 T = 10 mK H$_{0}$ = 51 A/m NbAu M(h) /  | M(h max )| h a) FIG. 6: a) the dependence of Φ ( T, h) on the magnetic field h. b) the relative free energy Ω (h) / Ω (h max ) (Φ (h max ) = Φ max ) as a function of the magnetic field h. FIG. 7: a) The magnetic moment of the NS structure versus the magnetic field h. b) Isothermal dc-magnetization curve at T = 7 mK for the sample 41AuNb [22]. Used by courtesy of A. C. Mota. Green-function technique [1] for the explanation of the experimental data (Mota et al) is a consequence of ac- count for states which are non-Andreev ones ( α > α$_{c}$ ) for the cylindrical NS structures. Geometrically, this can be seen from figure Fig. 1 . The quasiparticle tra- jectories ( α > α$_{c}$ ) hitting the dielectric boundary only are responsible for the paramagnetic current of a small amplitude ( ∼ 1 / ( k$_{F}$ · R )) (we neglected this current). The proximity effect is crucial for the reentrant effect. The amplitude of the resonance spikes in the density of states strongly depends on the probability of the Andreev reflection at the NS boundary. It is therefore assumed that the normal metal and the superconductor are in a good electric contact. The spectrum Eq. (4) was ob- tained by the method of multidimensional quasi-classical approach [15, 16]. In doing so, we assumed (i) the condi- tion of smallness of N-layer thickness in comparison with the radius of the cylindrical superconductor, (ii) the va- lidity of the model of a stepwise-varing order parameter of the structure, (iii) the independence of Δ on the magnetic field. This permitted us to pass over from the curved NS boundary to a flat one. The information about the cylin- drical geometry of the sample was retained because the critical angle at which a quasiparticle hits the dielectric boundary α is smaller than α$_{c}$ . The problem was further simplified by assuming that the reflected quasiparticle performed a reciprocating motion, i.e., a ”particle” and a ”hole” pass along the same trajectory but in opposite direction. Actually, here there exists a lot of quasi recip- rocating trajectories with the energies near ε ∼ 0. These trajectories lead to the spikes in the density of states and were taken into account for numerical computation. The