= Solution
{c}
In <Cartesian coordinates> on <Minkowski spacetime>, the <Conformal Killing equation> becomes
$$
\partial_\mu X_\nu+\partial_\nu X_\mu=2\lambda\eta_{\mu\nu}.
$$
Differentiate the equations with index pairs $(\nu,\rho)$, $(\mu,\rho)$ and $(\mu,\nu)$ by $\partial_\mu$, $\partial_\nu$ and $\partial_\rho$, respectively. Adding the first two and subtracting the third, <Clairaut's theorem> yields the <flat conformal Killing integrability> formula
$$
\partial_\mu\partial_\nu X_\rho
=\eta_{\nu\rho}\partial_\mu\lambda
+\eta_{\mu\rho}\partial_\nu\lambda
-\eta_{\mu\nu}\partial_\rho\lambda.
$$
Apply $\partial^\rho$ and use $\partial^\rho X_\rho=4\lambda$. With the <d'Alembert operator> $\Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta$, this gives
$$
4\partial_\mu\partial_\nu\lambda
=2\partial_\mu\partial_\nu\lambda-\eta_{\mu\nu}\Box\lambda,
\qquad
2\partial_\mu\partial_\nu\lambda=-\eta_{\mu\nu}\Box\lambda.
$$
A <tensor contraction> with $\eta^{\mu\nu}$ gives $2\Box\lambda=-4\Box\lambda$, so $\Box\lambda=0$. Returning to the preceding equation proves the stronger statement
$$
\boxed{\partial_\mu\partial_\nu\lambda=0.}
$$
Thus the <conformal Killing factor> is an <affine function>, not an arbitrary solution of the <wave equation>.
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