  The origin of massive field stars in the Large Magellanic Cloud (LMC) has
long been an enigma. The recent measurements of large offsets (~100 km/s)
between the heliocentric radial velocities of some very massive (O2-type) field
stars and the systemic LMC velocity provides a possible explanation of this
enigma and suggests that the field stars are runaway stars ejected from their
birth places at the very beginning of their parent cluster's dynamical
evolution. A straightforward way to prove this explanation is to measure the
proper motions of the field stars and to show that they are moving away from
one of the nearby star clusters or OB associations. This approach however is
complicated by the large distance to the LMC, which makes accurate proper
motion measurements difficult. We use an alternative approach for solving the
problem, based on the search for bow shocks produced by runaway stars. The
geometry of detected bow shocks would allow us to infer the direction of
stellar motion and thereby to determine their possible parent clusters. In this
paper we present the results of a search for bow shocks around six massive
field stars which were suggested in the literature as candidate runaway stars.
Using archival (Spitzer Space Telescope) data, we found a bow shock associated
with one of our program stars, the O2 V((f*)) star BI 237, which is the
first-ever detection of bow shocks in the LMC. Orientation of the bow shock
suggests that BI 237 was ejected from the OB association LH 82 (located at ~120
pc in projection from the star). A by-product of our search is the detection of
bow shocks generated by four OB stars in the field of the LMC and an arc-like
structure attached to the candidate luminous blue variable R81 (HD 269128). The
geometry of two of these bow shocks is consistent with the possibility that
their associated stars were ejected from the 30 Doradus star forming complex.
