Can we characterise concircular vector fields by their flow?

Question

Let $M$ be a Riemannian manifold. A vector field $X$ on $M$ is called concircular if $\nabla X=h \text{Id}_{TM}$ for some $h \in C^{\infty}(M)$, where $\nabla$ is the Levi-Civita connection. Every concircular field is conformal.

Can we characterize concircular fields by their flows alone?

I am looking for something analogous to the characterization of Killing,conformal, and divergence-free vector fields: Each of these creatures can be defined by some differential condition; $L_X g=0,L_X g=hg,\text{div} X=0$, or equivalently their flow is an isometry (conformal,volume-preserving).

The question is whether we can single out the concircular fields by only looking at their flow.

Here are some reasons to be pessimistic:

1. The concircular fields do not form a Lie-algebra (not closed under Lie bracket).

2. Consider the following "pointwise", finite-dimensional analogy between the covariant derivative $\nabla X$ of a conformal vector field, and elements in $T_{\text{id}}\text{CO}(n)$: A vector field is Killing if and only if $\nabla X$ is skew-symmetric, and is concircular if and only if $\nabla X$ is multiple of the identity.

In more detail, let $\text{O}(n),\text{CO}(n)$ be the groups of isometries and conformal linear transformations, respectively. Recall $T_{\text{id}}\text{O}(n)=\text{Skew}$ is the space of skew-symmetric matrices, and that $T_{\text{id}}\text{CO}(n)=\{ B \in M_n \, | \, B+B^T=\frac{2}{n}\text{trace}(B)\text{Id}\}$.

A field is Killing iff $\nabla X|_p \in \text{Skew}\big( \text{Hom}(T_pM,T_pM)\big)$. (This is analogous to $T_{\text{id}}\text{O}(n)=\text{Skew}$).

Define $W=\{ \lambda \cdot \text{id} \in M_n \, | \lambda \in \mathbb{R} \}$. Clearly $W \subseteq T_{\text{id}}\text{CO}(n)$, and in fact $T_{\text{id}}\text{CO}(n)=\text{Skew} \oplus W$.

Note that $W=T_{\text{id}}G$, where $G=\{ \lambda \cdot \text{id} | \lambda \neq 0\} \subseteq \text{CO}(n)$. This is the pointwise analog of $\nabla X=h\text{Id}_{TM}$. The problem is that while $\text{O}(n)$ has a manifold analog (the isometry group) which give rise to the Killing case, I don't see a clear analog in the manifold context of the group $G$. (Which conformal isometries correspond to multiples of the "identity"?).

Edit:

Here is one fact that distinguishes the concircular vector fields among the conformal vector fields; $\nabla_X X=hX $ is equivalent to the integral curves of $X$ being reparametrizations of geodesics. However, I am not sure every conformal vector field with this property is concircular.