Let \(A\) be a reduced rank 2 torsion-free abelian group. Denote by \(T\) the set of types of elements of \(A\), and by \(T\mathrm{'}\) the set of types of rank 1 torsion-free factor groups of A. The author characterizes such pairs \((T,T\mathrm{'})\), and counts the number of isomorphism and quasi-isomorphism classes of rank 2 groups realizing each such pair. A group is called completely anisotropic if no two independent elements have the same type. He characterizes the pairs \((T,T\mathrm{'})\), realizing completely anisotropic groups, and decides the number of isomorphism and quasi-isomorphism classes of completely anisotropic groups realizing such pairs.