Before revision: We give an O(k^3 n \polylog ( nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex {\em  capacities bounded by C, and k sources and sinks.
After revision: We give an O(k^3 n \polylog \log n \min(k,\log^2 n) \log^2( nC))-time algorithm for computing maximum integer flows in planar graphs with integer arc and vertex {\em capacities bounded by C, and k sources and sinks.