  The Standard Model (SM) represents our best description of the subatomic
world and it has been very successful in explaining how elementary particles
interact under the influence of the fundamental forces. Despite its far
reaching success in describing the building blocks of matter, the SM is still
incomplete; falling short to explain dark matter, baryogenesis, neutrino masses
and much more. The E36 experiment conducted at J-PARC in Japan, allows for
sensitivity to search for light $U(1)$ gauge bosons, in the muonic $K^+$ decay
channel. Such $U(1)$ bosons could be associated with dark matter or explain
established muon-related anomalies such as the muon $g_{\mu}-2$ value, and the
proton radius puzzle. A scintillating fiber target was used to stop a beam of
positively charged $K$ mesons. The $K^+$ products were detected with a
large-acceptance toroidal spectrometer capable of tracking charged particles
with high resolution, combined with a large solid angle CsI(TI) photon detector
and particle ID systems. A realistic simulation was employed to search for
these rare decays in the mass range of 20$-$100 MeV/$c^2$. Preliminary results
of the upper limits for the $A^\prime$ branching ratio $\mathcal{B}r(A^\prime)$
extracted at 95\% CL, will be discussed.
