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.