General solution of the one-phase Hele-Shaw problem for elliptic bubble

In the general case of several multipole we study one-phase Hele-Shaw flow with a moving boundary when surface tension effect is negligible. We find the explicit solution with nonstationary elliptic shape of the bubble for case when Hele-Shaw flow is produced by any combination of sink, dipole and quadropole at infinity. This general solution includes all known particular cases. In the particular case of a dipole at infinity we find new explicit solution with stationary elliptic shape of the bubble that is not symmetrical with respect to the flow.