An explicit representation for the axisymmetric solutions of the free Maxwell equations

17 pages. V4: version accepted for publication in Open Physics. A precision added: Eq (1) applies to totally propagating solutions. Sect. 3 that contains the main new results is unaffected. Three refs. added.International audienceGaray-Avendano & Zamboni-Rached (2014) defined two classes of axisymmetric solutions of the free Maxwell equations. We prove that the linear combinations of these two classes of solutions cover all totally propagating time-harmonic axisymmetric free Maxwell fields --- and hence, by summation on frequencies, all totally propagating axisymmetric free Maxwell fields. It provides an explicit representation for these fields. This will be important, e.g., to have the interstellar radiation field in a disc galaxy modelled as an exact solution of the free Maxwell equations