  We report on radio timing observations of the black widow binary pulsar
J0610$-$2100 and optical observations of its binary companion. The radio timing
observations extend the timing baseline to 16yr and reveal a marginal detection
of the orbital period derivative, but they show no significant evidence of
orbital variations such as those seen in other black widow pulsars.
Furthermore, no eclipses are seen in the observations at observing frequencies
ranging from 310 to 2700MHz. The optical VRI light curves were modulated with
the orbital period, reaching maximum brightness of $V=26.8$, $R=25.4$, and
$I=23.8$ at superior conjunction of the companion, confirming irradiation of
the companion by the pulsar. Modelling the light curves indicates that the
companion is likely not filling its Roche lobe, while having a moderate
inclination ($i > 54\degr$). We find an unusually low temperature and a low
irradiation for the irradiated hemisphere of the companion. We investigate the
absence of radio eclipses in PSR J0610-2100 and in other black widow systems in
relation to their binary, pulsar, and companion properties. We also discuss the
suitability of PSR J0610-2100 for pulsar timing array observations aimed at
detecting nano-Hertz gravitational waves.
