One of the two classes of non-diverging algebraically special left-flat spaces found by Fette, Janis and Newman is generalised by allowing the (self-dual part of the) Weyl tensor to become algebraically general. Penrose's conformal approach is used, thereby allowing the desired asymptotic behaviour of the solutions to be prescribed from the outset. The 'solutions' are, however, still subject to some 'reduced equations'. For a specific choice of initial data at conformal null infinity these reduced equations simplify to Plebanski's second heavenly equation.