We construct some ODE solutions for the fractional Yamabe problem in conformal geometry. The fractional curvature, a generalization of the usual scalar curvature, is defined from the conformal fractional Laplacian, which is a non-local operator constructed on the conformal infinity of a conformally compact Einstein manifold. These ODE solutions are a generalization of the usual Delaunay and, in particular, solve the fractional Yamabe problem with an isolated singularity at the origin $(-\Delta)^{\gamma} u= c_{n, {\gamma}}u^{\frac{n+2\gamma}{n-2\gamma}}, u>0.$ This is a fractional order ODE for which new tools need to be developed. The key of the proof is the computation of the fractional Laplacian in polar coordinates.