Analytic continuation of the kite family

We consider results for the master integrals of the kite family, given in terms of ELi-functions which are power series in the nome $q$ of an elliptic curve. The analytic continuation of these results beyond the Euclidean region is reduced to the analytic continuation of the two period integrals which define $q\mathrm{.}$ We discuss the solution to the latter problem from the perspective of the Picard-Lefschetz formula.