= Solution
Because the <Levi-Civita connection> preserves the spacetime volume form, only derivatives of $X$ remain in $\mathcal L_XT$. Lower the raised indices in $T_{ab}{}^{cd}=\epsilon_{ab}{}^{cd}$, contract successively with the supplied products of two Levi-Civita tensors, and separate $\nabla_aX_b$ into antisymmetric, trace and symmetric trace-free parts. The antisymmetric part cancels automatically, while $\mathcal L_XT=0$ is equivalent to vanishing of the symmetric trace-free part:
$$
\nabla_{(a}X_{b)}-\frac14g_{ab}\nabla_cX^c=0.
$$
Therefore
$$
\boxed{\nabla_aX_b+\nabla_bX_a
=\frac12g_{ab}\nabla_cX^c},
$$
so $\boxed{\alpha=1/2}$. This is the four-dimensional <Conformal Killing equation>, and $X$ is a <Conformal Killing vector field>.
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