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A homogeneous Riemannian manifold $M=G/H$ is called a ``g.o. space'' if every geodesic on $M$ arises as an orbit of a one-parameter subgroup of $G$. Let $M=G/H$ be such a ``g.o. space'', and $m$ an $\text{Ad}(H)$-invariant vector subspace of $\text{Lie}(G)$ such that $\text{Lie}(G)=m\oplus\text{Lie}(H)$. A {\sl geodesic graph} is a map $\xi:m\to\text{Lie}(H)$ such that $$ t\mapsto \exp(t(X+\xi(X)))(eH) $$ is a geodesic for every $X\in m\setminus\{0\}$. The author calculates explicitly such geodesic graphs for certain special 2-step nilpotent Lie groups. More precisely, he deals with ``generalized Heisenberg groups'' (also known as ``H-type groups'') whose center has dimension not exceeding three.
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