  We study binary systems in which a stellar mass compact object spirals into a
massive black hole, known as extreme mass ratio inspirals, in scenarios with a
new fundamental scalar field. Earlier work has shown that, in most interesting
such scenarios and to leading order in the mass ratio, the massive black holes
can be adequately approximated by the Kerr metric and the imprint of the scalar
field on the waveform is fully controlled by the scalar charge of the stellar
mass object. Here we use this drastic simplification in the inspiral modelling
and consider eccentric equatorial orbits. We study how the scalar charge
affects the orbital evolution for different eccentricities and different values
of the black hole spin. We then determine how changes in the orbital evolution
get imprinted on the waveform and assess LISA's capability to detect or
constrain the scalar charge.
