  Symplectic and symmetry analysis for studying MHD superfluid flows is
devised, a new version of the Z. Peradzynski helicity theorem based on
differential - geometric and group-theoretical methods is derived. Having
reanalyzed the Peradzynski helicity theorem within the modern symplectic theory
of differential- geometric structures on manifolds, a new unified proof and a
new generalization of this theorem for the case of compressible MHD superfluid
flow are proposed. As a by-product, a sequence of nontrivial helicity type
local and global conservation laws for the case of incompressible superfluid
flow, playing a crucial role for studying the stability problem under suitable
boundary conditions, is constructed.
