  We solve the Einstein constraint equations for a first-order causal viscous
relativistic hydrodynamic theory in the case of a conformal fluid. For such a
theory, a direct application of the conformal method does not lead to a
decoupling of the equations, even for constant-mean curvature initial data. We
combine the conformal method applied to a background perfect fluid theory with
a perturbative argument in order to obtain the result.
