The evolution of scroll rings with filaments of arbitrary
    shape has been studied analytically for reaction-diffusion
    systems with all the state variables having the same
    diffusion coefficients. Belonging to this class of systems
    are , in particular, models of the Belousov-Zhabotinskii(BZ)
    chemical reaction. It has been found that the speed of
    decrease of an area bounded by the planar scroll filament is
    given by S = - 2..D (D is the diffusion coefficient) and is
    independent of the kind of model used to describe an active
    medium, time and filament shape;i.e. the integral ...v ds ==
    -2..D, where ds is the differential of the filament length
    and v is the drift velocity, always, always remains valid.
    Using this integral invariant a number of problems on the
    drift of closed and unclosed three-dimensional vortices have
    been solved. The theoretical predictions have been verified
    in experiments with the BZ reaction.