A Monte Carlo event generator is constructed for a two-Higgs-doublet model with maximal CP symmetry, the MCPM. The model contains five physical Higgs bosons; the $rho$, behaving similarly to the standard-model Higgs boson, two extra neutral bosons $h$ and $h$, and a charged pair $H^pm$. The special feature of the MCPM is that, concerning the Yukawa couplings, the bosons $h$, $h$ and $H^pm$ couple directly only to the second generation fermions but with strengths given by the third-generation-fermion masses. Our event generator allows the simulation of the Drell-Yan-type production processes of $h$, $h$ and $H^pm$ in proton-proton collisions at LHC energies. Also the subsequent leptonic decays of these bosons into the $mu^+ mu^-$, $mu^+ u_mu$ and $mu^- bar u_mu$ channels are studied as well as the dominant background processes. We estimate the integrated luminosities needed in $p p$ collisions at center-of-mass energies of 8 TeV and 14 TeV for significant observations of the Higgs bosons $h$, $h$ and $H^pm$ in these muonic channels.
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