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Abstract Geodesic equations are solved when at least two of $$\theta $$ θ , $$\phi $$ ϕ and $$\psi $$ ψ are constant, or r is constant, on scalar flat metrics of Eguchi–Hanson type. They can also be solved also on Eguchi–Hanson metrics which are Ricci flat if only $$\phi $$ ϕ is constant. However, the explicit solution of the geodesic equations is not available yet if only $$\psi $$ ψ is constant.