On the Law of Transformation of Affine Connection and its Integration.   Part 1. Generalization of the Lame equations
  The law of transformation of affine connection for n-dimensional manifolds as the system of nonlinear equations on local coordinates of manifold is considered. The extension of the Darboux-Lame system of equations to the spaces of constant negative curvature is demonstrated. Geodesic deviation equation as well as the equations of geodesics are presented in the form of the matrix Darboux-Lame system of equations. 
Valery S. Dryuma