We show that the Coulomb blockade in parallel dots pierced by magnetic flux $Phi$ completely blocks the resonant current for any value of $Phi$ except for integer multiples of the flux quantum $Phi_0$. This non-analytic (switching) dependence of the current on $Phi$ arises only when the dot states that carry the current are of the same energy. The time needed to reach the steady state, however, diverges when $Phito nPhi_0$.
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