# Symplectic gradients whose span doesn't intersect Lie group orbits

I asked a similar question some hours ago. But thinking about my problem, I found a loophole in my arguments, so that my question wasn't the right one. This one is what I wanted to ask:

Let $G$ be a compact Lie group with some left-invariant Riemannian metric $ds$. Therefore we get a symplectic form $\omega$ on $TG$, if we pullback the canonical symplectic form on $T^*G$ by the Riemannian metric.

We are considering now $G$ and $ds$, such that there are functions $f_1, \dots, f_k \colon TG \to \mathbb{R}$, which are invariant under the geodesic flow, which mutually Poisson-commute and which are invariant under some Lie subgroup $K \subset G$. The number $k$ of my functions is at least $\dim G - \dim K$.

Denote by $X_{f_j}\in \mathfrak{X}(TG)$ the symplectic gradient of $f_j$. Since the functions $f_j$ are $K$-invariant, we know that $X_{f_j}(v) \in \left(T_v\left(K \cdot v\right)\right)^\omega$ for almost all $v \in TG$.

Are there some known conditions, so that we find such functions $f_1, \dots, f_k$, such that the linear span of the vectorfields $X_{f_1}, \dots, X_{f_r}$ intersects the $K$-orbits only in $0$ in some open and dense subset of $TG$? That means, that $$\operatorname{span}\left\{X_{f_1}(v), \dots, X_{f_k}(v)\right\} \cap T_v(K \cdot v) = \{0\}$$ for almost all $v \in TG$.

Therefore these functions satisfy

$$\operatorname{span}\{X_{f_1}(v), \dots, X_{f_k}(v) \} \subset \left(\left(T_v\left(K \cdot v\right)\right)^\omega \setminus T_v\left(K \cdot v\right)\right).$$

How could I construct such Lie groups $K \subset G$, with left-invariant metric $ds$?

Clearly one necessary condition is, that the $K$-orbits aren't coisotropic submanifolds of $TG$. But since $\dim K < \dim G = \frac{1}{2}\dim TG$, the $K$-orbits are never coisotropic in $TG$.

Edit: As I suppose that there are only very deep conditions (if there are some at all), I'd be happy to know, if there are some examples. For instance if $(G,K) = \left(U(n),U(r) \right)$ would satisfy this. Futhermore, would it be possible to construct examples if the metric on $G$ were not $G$-invariant, but only $K$-invariant?

Although this is not quite the same setting, the question arises from the following fact:

Having a complete surface of revolution $(N,g)$, with metric $g$, the geodesic flow is completely integrable in $TN$, since we can take Clairaut's first integral and $f(v) = g(v,v)$ for $v \in TN$. Now it is easy to show that almost everywhere the symplectic gradient $X_f$ is not tangent to the $S^1$-orbits in $TN$.