  (Abridged) - The combination of several mass-decreasing processes may
critically affect the structure of open clusters (OCs), to the point that most
dissolve into the field in a time-scale shorter than $\approx1$ Gyr. In this
paper we focus on 4 candidate old star clusters, namely, LK 1, LK 10, FSR 1521,
and FSR 1555. To analyse the target clusters we construct near infrared
colour-magnitude diagrams (CMDs) and derive stellar radial density profiles
(RDPs). The CMDs are constructed using 2MASS \jj, \hh, and \ks bands, and the
intrinsic morphologies of the target OCs within these diagrams are revealed by
applying a field-star decontamination algorithm. Fundamental parameters are
estimated with Padova isochrones built for the 2MASS filters. We derive
extinctions to the objects within the range $3.4\le\aV\le8.9$, which makes them
suitable for the near-infrared analysis, ages within $1.0 - 2.0$ Gyr, and
distances from the Sun within $1.4 - 4.5$ kpc. These distances, in conjunction
with the positions in the sky, place the present 4 OCs close to the solar
circle ($\la0.6$ kpc). For LK 10 our photometry reaches a depth $\approx3$ mags
below the main sequence turn off, from which we derive a relatively steep mass
function slope ($\chi=2.4\pm0.4$) when compared to the Salpeter value
($\chi=1.35$). LK 10 is a rather massive old OC, with a mass within $1360\le
m(\ms)\le4400$, for stars in the observed magnitude range and the extrapolation
to $0.08 \ms$, respectively. The mass estimated in the restricted magnitude
range for the remaining, more distant OCs is within $260\le m(\ms)\le380$.
However, similarity with the CMD morphology and red clump of LK 10 suggests
that they may be as massive as LK 10.
