We report measurements of the branching fractions of singly Cabibbo-suppressed decays $Lambda_c^+ to p eta$ and $Lambda_c^+ to p pi^0$ using the full Belle data sample corresponding to an integrated luminosity of 980.6 $rm fb^{-1}$. The data were collected by the Belle detector at the KEKB $e^{+}$$e^{-}$ asymmetric-energy collider. A clear $Lambda_c^+$ signal is seen in the invariant mass distribution of $p eta$. The signal yield of the $Lambda_c^+ to p eta$ process is $7734 pm 263$; from this, we measure the ratio of branching fractions ${cal B}(Lambda_c^+ to p eta)/{cal B}(Lambda_c^+ to p K^- pi^+) = (2.258 pm 0. 077(rm stat. ) pm 0.136(rm syst. ))times 10^{-2}$, from which we infer the branching fraction ${cal B}(Lambda_c^+ to p eta) = (1.42 pm 0.05(rm stat.) pm 0.11(rm syst.)) times 10^{-3}$. In addition, no significant signal for $Lambda_c^+ to p pi^0$ is found so an upper limit on the branching fraction of ${cal B}(Lambda_c^+ to p pi^0)<8.0 times 10^{-5}$ at 90% credibility level is set, more than three times better than the best current upper limit.
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