The BFKL pomeron describes high energy scattering in perturbative gauge theory in the Regge limit. In this talk I will describe some motivation for studying BFKL using the techniques of integrability, as well as some significant difficulties encountered in the process. In order to overcome some of these problems, I will describe new analyticity conditions for solving the relevant Baxter equation which lead to a solution which works directly also in the BFKL regime. These conditions may be a step towards an exact formulation valid at any coupling, which however remains as an outstanding open problem.