  The relativistic dynamic equations are derived for a
superfluid-superconducting mixture coupled to the electromagnetic field. For
definiteness, and bearing in mind possible applications of our results to
neutron stars, it is assumed that the mixture is composed of superfluid
neutrons, superconducting protons, and normal electrons. Proton
superconductivity of both I and II types is analysed, and possible presence of
neutron and proton vortices (or magnetic domains in the case of type-I proton
superconductivity) is allowed for. The derived equations neglect all
dissipative effects except for the mutual friction dissipation and are valid
for arbitrary temperatures (i.e. they do not imply that all nucleons are
paired), which is especially important for magnetar conditions. It is
demonstrated that these general equations can be substantially simplified for
typical neutron stars, for which a kind of magnetohydrodynamic approximation is
justified. Our results are compared to the nonrelativistic formulations
existing in the literature and a number of discrepancies are found. In
particular, it is shown that, generally, the electric displacement ${\pmb D}$
does not coincide with the electric field ${\pmb E}$, contrary to what is
stated in the previous works. The relativistic framework developed here is
easily extendable to account for more sophisticated microphysics models and it
provides the necessary basis for realistic modelling of neutron stars.
