  We present equilibrium models of relativistic magnetised, infinite,
axisymmetric jets with rotation propagating through an homogeneous,
unmagnetised ambient medium at rest. The jet models are characterised by six
functions defining the radial profiles of density, pressure, and the toroidal
and axial components of velocity and magnetic field. Fixing the ambient
pressure and the jet rest-mass density and axial components of the flow
velocity and magnetic field, we analyze the influence of the toroidal magnetic
field and several rotation laws on the structure of the equilibrium models. Our
approach excludes by construction the analysis of the self-consistently
magnetically launched jet models or the force-free equilibrium solutions.
Several forbidden regions in the magnetic pitch angle/magnetization plane are
found where models of the class considered in our study could not be settled.
These forbidden regions are associated with the existence of maximum axial and
toroidal magnetic field components compatible with the prescribed equilibrium
condition at the jet surface, and/or an excess of centrifugal force producing
gaps with negative pressures in the jet. The present study can be easily
extended to jet models with different transversal profiles and magnetic field
configurations.
  In the last part of the paper, we test the ability of our RMHD code to
maintain steady equilibrium models of axisymmetric RMHD jets in one and two
spatial dimensions. The one dimensional numerical simulations serve also as a
consistency proof of the fidelity of the analytical steady solutions discussed
in the first part of the paper. The present study allows us to build initial
equilibrium jet models with selected properties for dynamical (and emission)
simulations of magnetised relativistic jets with rotation.
