Complete integration of the aligned Newman Tamburino Maxwell solutions

In a previous work we examined the generalization of the Newman Tamburino metrics in the presence of an aligned Maxwell field [1]. It was shown there that the so called ‘spherical’ class did not admit any solutions and that consistency of the field equations therefore required the ‘cylindrical’ condition |ρ| = |σ|. The existence of solutions in the cylindrical class was left however as an open question. In the present paper we obtain a complete integration of the Einstein Maxwell field equations for this problem. We first show that the Newman-Penrose, Bianchi and Maxwell equations form an integrable system and then proceed to integrate the first Cartan structure equations. After obtaining the general solution we look at the limiting case, where the charge Q of the Maxwell field goes to zero. Surprisingly in this limit, we do not recover the empty space metric obtained by Newman and Tamburino, but rather the special case admitting two Killing vectors. In §2 we describe the choice of tetrad and the integration method. The resulting metric is presented at the end of this section. In §3 we discuss the vacuum limit of the metric and its symmetry properties.