The gradient flow in a least squares problem on a Lie group takes a Lax form . We associate the Lax equation with homogeneous spaces and symmetric spaces of compact simple Lie groups. The critical points of the Lax equation lie in the Cartan subalgebras of the simple Lie algebras. A reduction from homogeneous spaces to symmetric spaces is described by a ‘coalescence’ of roots. For the complex Grassmann manifold, it is shown that an initial value problem of the Lax equation can be uniquely solved. Some applications to a least squares fitting problem and a linear programming problem are discussed.