Embedded spheres in S2×S1#⋯#S2×S1

We give an algorithm to decide which elements of π2(#kS2×S1) can be represented by embedded spheres up to conjugacy. Such spheres correspond to splittings of the free group Fk on k generators. We also give an algorithm to decide when classes in π2(#kS2×S1) can be represented by disjoint embedded spheres.Our methods may be useful in studying the splitting complex of a free group, and hence the group of outer automorphisms.