Polynomial-exponential equations - Some new cases of solvability

Recently, Brownawell and the second author proved a 'non-degenerate' case of the (unproved) 'Zilber Nullstellensatz' in connexion with 'Strong Exponential Closure'. Here, we treat some significant new cases. In particular, these settle completely the problem of solving polynomial-exponential equations in two complex variables. The methods of proof are also new, as is the consequence, for example, that there are infinitely many complex z$z$ with ez+e1/z=1$e<^>z+e<^>{1/z}=1$.