Non-thermal radiation has been predicted within bow shocks around runaway stars by recent theoretical works. We present X-ray observations towards the runaway stars $zeta$ Oph (Chandra and Suzaku) and BD+43$^{circ}$3654 (XMM-Newton) to search for the presence of non-thermal X-ray emission. We found no evidence of non-thermal emission spatially coincident with the bow shocks, nonetheless, diffuse emission is detected in the vicinity of $zeta$ Oph. After a careful analysis of its spectral characteristics we conclude that this emission has a thermal nature with a plasma temperature of $T approx 2 times10^{6}$ K. The cometary shape of this emission seems to be in line with recent predictions of radiation-hydrodynamic models of runaway stars. The case of BD+43$^{circ}$3654 is puzzling as non-thermal emission has been reported in a previous work for this source.
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