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We study the structure of CSC-1 reducible conformal metrics on a compactRiemann surface with finite conical singularities. We prove that any compactRiemann surface with a CSC-1 reducible conformal metric of finite conicalsingularities can be divided into a finite number of pieces by cutting alonggeodesics where each piece is isometric to some football. This allows us tostudy the existence of CSC-1 reducible conformal metrics with finite conicalsingularities on a compact Riemann surface. As an application, we give an anglecondition of the existence of CSC-1 reducible conformal metrics with finiteconical singularities on a compact Riemann surface.

The structure of CSC-1 reducible conformal metrics on a compact Riemann surface with finite conical singularities