Abstract
We find a class of twisted and differentially rotating neutron star magnetospheres that do not have a light cylinder, generate no wind and thus do not spin-down. The magnetosphere is composed of embedded differentially rotating flux surfaces, with the angular velocity decreasing as (equivalently, becoming smaller at the foot-points closer to the axis of rotation). For each given North-South self-similar twist profile there is a set of self-similar angular velocity profiles (limited from above) with a "smooth", dipolar-like magnetic field structure extending to infinity. For spin parameters larger than some critical value, the light cylinder appears, magnetosphere opens up, and the wind is generated.
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