the magnetic flux inside the loops.](Fig6)

The pair velocity may increase up to the depairing velocity $|v| \leq v _{c} = \hbar /m \surd 3 \xi (T)$ in narrow loop segments [@Tink75] without a change of the quantum number $n$, Fig.5. The quantum number can change when a density of Cooper pairs diminishes for a while from $n_{s} = 2n_{s0}/3$ [@Tink75] to $n_{s} = 0$ at $|v| \geq  v _{c} $ in a segment, where $n_{s0}$ is the density at $|v| = 0$. The quantum number of homogeneous ring changes at $|n - \Phi /\Phi _{0}| \approx r/\surd 3 \xi (T)$, Fig.5, when the velocity increases only because of magnetic flux $\Phi \approx BS$ change in accordance with (2). The jump of the velocity (2) and the persistent current, observed in this case [@Geim2003; @Moler2007; @Nat2016], cannot be used for a magnetometer designing because of the problem with a hysteretic field characteristic. The quantum number can change from $n$ to $n+1$ at a definite magnetic flux $\Phi = (n+0.5) \Phi _{0}$, Fig.5, due to the switching of a segment in the normal state for a while by an external current at $T < T _{c}$ [@PRB2014C], by thermal fluctuations at 