We report on the search for steady point-like sources of neutral particles around 10$^{18}$ eV between 2008 May and 2013 May with the scintillator surface detector of the Telescope Array experiment. We found overall no significant point-like excess above 0.5 EeV in the northern sky. Subsequently, we also searched for coincidence with the Fermi bright Galactic sources. No significant coincidence was found within the statistical uncertainty. Hence, we set an upper limit on the neutron flux that corresponds to an averaged flux of 0.07 km$^{-2}$ yr$^{-1}$ for $E>1$ EeV in the northern sky at the 95% confidence level. This is the most stringent flux upper limit in a northern sky survey assuming point-like sources. The upper limit at the 95% confidence level on the neutron flux from Cygnus X-3 is also set to 0.2 km$^{-2}$ yr$^{-1}$ for $E>0.5$ EeV. This is an order of magnitude lower than previous flux measurements.
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