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We show that for every second order Fuchsian linear differential equation $E$with $n$ singularities of which $n-3$ are apparent there exists ahypergeometric equation $H$ and a linear differential operator with polynomialcoefficients which maps the space of solutions of $H$ into the space ofsolutions of $E$. This map is surjective for generic parameters. This justifiesone statement of Klein (1905). We also count the number of such equations $E$with prescribed singularities and exponents. We apply these results to thedescription of conformal metrics of curvature $1$ on the punctured sphere withconic singularities, all but three of them having integer angles.

Fuchsian Equations with Three Non-Apparent Singularities