@Soham: Well, that's a good question. Certainly, the Legendrian isotopy class of the link is an invariant, but it is not clear to me that there is anything beyond that. In some sense, you are asking whether the 'natural' radial vector field on $CT$ can be extended to a conformally symplectic vector field on a neighborhood of the singular point, or something like that. I'll think about whether that can be made precise. In any case, Balarka Sen's cautionary counterexample for $n=1$ would have to be dealt with, maybe by requiring that $C$ be connected.

@PaulCusson: No, that equation is what I meant. It is true that what I wrote is equivalent to $$\alpha^2_{21}+\alpha^3_{31}=\alpha^1_{12}+\alpha^3_{32}=\alpha^1_{13}+\alpha^2_{23}=0,$$ but that is what I intended. This is the condition that, when $\xi^i$ are the dual 1-forms to the $X_i$, the $2$-forms $\xi^i\wedge\xi^j$ should be closed for every pair $i\not=j$. This is a natural condition on the frame field $X_i$.

OK. As I suspected, the three-dimensional examples easily generalize to all higher dimensions as a 3-parameter family of non-trivial gradient almost Ricci solitons with harmonic curvature. They are not completely explicit, though, because each equivalence class of solutions corresponds to an integral curve of a vector field in $\mathbb{R}^3$. That vector field depends on a parameter: the (constant) scalar curvature $S=n(n{-}1)c$ of the metric $g$. I don't know how to integrate the vector field in elementary terms, but in the case $c=0$ phase portrait methods give good qualitative information.

I thought a little bit about this and did a few caculations. In dimension 3, at least, it is not true that a gradient almost Ricci soliton with harmonic curvature is locally Einstein. There is a $3$-parameter family of mutually non-isometric, nontrivial examples $(g,f,\lambda)$ in dimension $3$ with the function $\lambda$ and the sectional curvatures not being constant. I haven't looked at higher dimensions, but I don't see why it would necessarily fail there.

Thanks! I could imagine how to prove the special case of $\mathrm{SU}(3)/\mathbb{T}^2$, but the details were messy. It's good to know that it works for all $G/T$, which I wouldn't even have thought of attempting to prove.

@IanGershonTeixeira: There's no need to assume 'transitive', so I didn't assume that. If a connected, compact group $G$ acts effectively on $\mathbb{CP}^n$ then the dimension of $G$ is at most $n(n{+}2)$. If equality holds, then the action is transitive and $G$ is the quotient of $\mathrm{SU}(n{+}1)$ by its center. As to the final question in your comment, the answer is 'no'. The 10-dimensional group $\mathrm{Sp}(2)$ acts transitively on $\mathbb{CP}^3$, and the action is almost effective, the center (isomorphic to $\mathbb{Z}_2$) acts trivially. Similar results for any $\mathbb{CP}^{2n+1}$.

@RamiroLafuente: Of course you are right. That remark was wrong-headed. I'll remove it. Anyway, I have realized that there is a much simpler argument for the maximum symmetry of $\mathbb{CP}^n$ that works for all $n$, so I'll replace that entire segment.