= Solution
The complex form of the <Chordal Loewner equation>, driven by a continuous real <Loewner driving function>, is
$$
\boxed{\partial_tg_t(z)=\frac2{g_t(z)-\xi_t},
\qquad g_0(z)=z.}
$$
For $z\in\mathbb C\setminus\{\xi_0\}$ this is an <ordinary differential equation> up to its maximal lifetime, when the solution reaches the driving singularity. The coefficients respect complex conjugation, so the lower-half-plane flow is the conjugate of the upper-half-plane flow. On surviving real points it is the real boundary flow. The <hydrodynamic normalization at infinity> and factor two correspond to <half-plane-capacity parameterization> $\operatorname{hcap}(K_t)=2t$.
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