We report an observation of the $B^{pm} to J/psi eta K^{pm}$ and $B^0 to J/psi eta K^0_S$ decays using 772$times 10^{6}$ $Bbar{B}$ pairs collected at the $Upsilon(4S)$ resonance with the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. We obtain the branching fractions ${cal B}(B^{pm}rightarrow J/psieta K^{pm})=(1.27pm 0.11{rm (stat.)pm 0.11{rm (syst.)})}times10^{-4}$ and ${cal B}(B^0to J/psi eta K^0_S)=(5.22 pm 0.78 {rm(stat.)} pm 0.49{rm(syst.)})times10^{-5}$. We search for a new narrow charmonium(-like) state $X$ in the $J/psi eta$ mass spectrum and find no significant excess. We set upper limits on the product of branching fractions, ${cal B}(B^pm to XK^pm){cal B}(X to J/psi eta)$, at 3872 MeV$/c^2$ where a $C$-odd partner of X(3872) may exist, at $psi(4040)$ and $psi(4160)$ assuming their known mass and width, and over a range from 3.8 to 4.8 GeV$/c^2$. % at a 5 MeV$/c^2$ step. The obtained upper limits at 90% confidence level for $X^{C{rm -odd}}(3872)$, $psi(4040)$ and $psi(4160)$ are 3.8$times 10^{-6}$, 15.5$times 10^{-6}$ and 7.4$times 10^{-6}$, respectively.
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