  In models with extended Higgs sectors, Higgs-boson-mediated Lepton Flavour
Violation (LFV) can naturally appear. We study the physics potential of an
electron-photon collider %as an option of the International Linear Collider on
searching LFV processes $e^-\gamma\to\ell^-\varphi (\ell=\mu,\tau; \varphi=H,
A)$ where $H$ and $A$ are extra CP even and odd Higgs bosons, respectively, in
the minimal supersymmetric standard model and the effective two Higgs doublet
model. The production cross section can be significantly large for the maximal
allowed values of the LFV coupling constants under the current experimental
data. Present experimental upper bounds on the effective LFV coupling constants
would be considerably improved by searching these processes, which would be
better than MEG and COMET experiments and also those at LHCb and SuperKEKB.
Moreover, one can separately measure chirality of effective LFV coupling
constants via these processes by selecting electron polarizations.
