I have been investigating conformal Killing vectors on pseudo-Riemannian manifolds, that is, vectors which obey $$ \mathcal{L}_X g = \lambda g$$ where $g$ is the metric and $\lambda$ is some function. In index form, this is $$\nabla_\mu X_\nu + \nabla_\mu X_\nu = \frac{2}{n}\nabla_\rho X^\rho g_{\mu\nu},$$ where $n$ is the dimension of the manifold and we have taken the trace to work out $\lambda$.
I am particularly interested in conformal Killing vectors which also obey the following constraint $$ \Box\nabla_\mu X^\mu = 0,$$ where $\Box = \nabla_\mu \nabla^\mu$ is the Laplacian. This constraint is equivalent to requiring that $\lambda$ above be harmonic. The set of conformal Killing vectors obeying this constraint is closed under Lie brackets and thus forms an algebra. They also have some interesting properties, e.g., $$R \nabla_\mu X^\mu = -\frac{n}{2} X^\mu \nabla_\mu R,$$ where $R$ is the Ricci scalar.
My question is simple. Do these vectors have a name? If so, what else is known about them? If not, what would be a good name for them?