Differentiating the spherical-geodesic first integral gives
(1)
The same equation follows from the affine geodesic equation after eliminating the arc-length derivatives. The squared first-derivative equation also fixes the normalization constant; a second-order solution must retain that first integral.

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  1. Spherical-geodesic first integral
  2. Clairaut first integral for a surface of revolution
  3. Surface of revolution
  4. Differential geometry
  5. Geometry and topology
  6. Area of mathematics
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