Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (**Edit**: which are sub-bundles of $\Omega^*(M, \mathfrak{g})$, namely forms valuded in Lie-algebra $\mathfrak{g}$) over $M$ with the same rank. Denote $\tilde \Gamma(E)$ and $\tilde \Gamma(F)$ the sections of $E$ and $F$ that are invariant under $R$, namely, for instance

$\mathcal{L}_R \tilde \sigma_{E} = 0$.

Let $D$ be some differential operator $D: \tilde \Gamma(E) \to \tilde \Gamma(F)$, which obviously commutes with $\mathcal{L}_R$.

My target $\dim{\rm ker}D - \dim \ker D^*$. What is the corresponding framework to do this? What is the property of $D$ for this quantity to be well defined? Any reference will be appreciated!(I have basic knowledge about usual index-theorem for elliptic complex)

Thanks.