Let T and T′ be sets of
types. This paper describes necessary and sufficient conditions on (T,T′) for the
existence of a rank 2 torsion-free abelian group A such that T is the set of types of
elements of A, and T′ is the set of types of rank 1 factor groups of A. Moreover, it
classifies all such A and gives necessary and sufficient conditions for A to be
completely anisotropic.