I am looking for an equation analogous to the Euler-Poincare equations for a right invariant Finlser metric except I want the geodesics which are parallel to a linear affine distribution on $SU(n)$. Additionally, the distribution is defined by right invariant vector fields.

Due to the abundance of right invariance, I am hoping for an equation which holds only in $\mathfrak{su}(n)$, for the generator to such a geodesic. I have attempted to use Lagrange multipliers to achieve what I want, but I am confused about how exactly to do this at the algebra level.

affine linear' distribution, do you mean that you require that the tangent vectors actuallylie ina right-invariant subset of the tangent bundle that, at the identity, is an affine subspace of the tangent space at the identity? I'm sorry to be so picky about the language, but I want to be sure that I understand what you mean byaffine linear distribution, which, if I am right, is what most people call anaffine distribution. $\endgroup$