Haken spheres in genus 2 Heegaard splittings of nonprime 3-manifolds

A Haken sphere S in a 3-manifold M with a Heegaard splitting V∪FW is a separating 2-sphere in M which intersects the Heegaard surface F only in an essential circle in F. It is known how any two Haken spheres in the genus 2 Heegaard splitting of S3 or a connected sum of two lens spaces are related. In the present paper, we show how any two Haken spheres in a genus 2 Heegaard splitting of S2×S1 # S2×S1 or S2×S1 # L(p,q) are related, where L(p,q) is a lens space.