  Non-radiative solutions of energy critical wave equations are such that their
energy in an exterior region $|x|>R+|t|$ vanishes asymptotically in both time
directions. This notion, introduced by Duyckaerts, Kenig and Merle (J. Eur.
Math. Soc., 2011), has been key in solving the soliton resolution conjecture
for these equations in the radial case. In the present paper, we first classify
their asymptotic behaviour at infinity, showing that they correspond to a
$k$-parameters family of solutions where $k$ depends on the dimension. This
generalises the previous results (Duyckaerts, Kenig and Merle, Camb. J. Math.,
2013 and Duyckaerts, Kenig, Martel and Merle, Comm. Math. Phys., 2022) in three
and four dimensions. We then establish a unique maximal extension of these
solutions.
