  We demonstrate that, in contrast to the single-component Abrikosov vortex, in
two-component superconductors vortex solutions with exponentially screened
magnetic field exist only in exceptional cases: in the case of vortices
carrying an integer number of flux quanta, and in a special parameter limit for
half-quantum vortices. For all other parameters the vortex solutions have
delocalized magnetic field with a slowly decaying tail. Furthermore, we
demonstrate a new effect which is generic in two-component systems but has no
counterpart in single-component systems: on exactly half of the parameter space
of the $U(1)\times U(1)$ Ginzburg-Landau model, the magnetic field of a generic
fractional vortex inverts its direction at a certain distance from the vortex
core.
