I'll present a recent joint work with A. Buryak and D. Zvonkine, where we study the moduli spaces of residueless meromorphic differentials, i.e., the closures, in the moduli space of stable curves, of the loci of smooth curves whose marked points are the zeros and poles of prescribed orders of a meromorphic differential with vanishing residues. Our main result is that intersection theory on these spaces is controlled by an integrable system containing the celebrated Kadomtsev-Petviashvili (KP) hierarchy as a reduction to the case of differentials with exactly two zeros and any number of poles. This fact has several deep consequences and in particular it relates the aforementioned moduli spaces with Hurwitz theory, representation theory of s/2(C), integrability and a conjecture of Schmitt and Zvonkine on the r=0 limit of Witten's r-spin classes.