  Isolated photons with high transverse energy have been studied in deep
inelastic $ep$ scattering with the ZEUS detector at HERA, using an integrated
luminosity of $326\,$ pb$^{-1}$ in the range of exchanged-photon virtuality $10
- 350$ GeV$^2$. Outgoing isolated photons with transverse energy $4<E_T^\gamma<
15$ GeV and pseudorapidity $-0.7 <\eta^\gamma< 0.9$ were measured with
accompanying jets having transverse energy and pseudorapidity $2.5
<E_T^{jet}<35$ GeV and $-1.5<\eta^{jet}< 1.8$, respectively. Differential cross
sections are presented for the following variables: the fraction of the
incoming photon energy and momentum that is transferred to the outgoing photon
and the leading jet; the fraction of the incoming proton energy transferred to
the photon and leading jet; the differences in azimuthal angle and
pseudorapidity between the outgoing photon and the leading jet and between the
outgoing photon and the scattered electron. Comparisons are made with
theoretical predictions: a leading-logarithm Monte Carlo simulation, a
next-to-leading-order QCD prediction, and a prediction using the
$k_T$-factorisation approach.
