We report branching fraction measurements of four decay modes of the $Lambda_{c}^{+}$ baryon, each of which includes an $eta$ meson and a $Lambda$ baryon in the final state, and all of which are measured relative to the $Lambda_{c}^{+} rightarrow p K^{-} pi^{+}$ decay mode. The results are based on a $980~mathrm{fb^{-1}}$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider. Two decays, $eta Sigma^{0} pi^{+}$ and $Lambda(1670) pi^{+}$, are observed for the first time, while the measurements of the other decay modes, $Lambda_{c}^{+} rightarrow etaLambdapi^{+}$ and $etaSigma(1385)^{+}$, are more precise than those made previously. We obtain $mathcal{B}(Lambda_{c}^{+} rightarrow eta Lambda pi^{+})/mathcal{B}(Lambda_{c}^{+} rightarrow p K^{-} pi^{+})$ = $0.293 pm 0.003 pm 0.014$, $mathcal{B}(Lambda_{c}^{+} rightarrow eta Sigma^{0} pi^{+})/mathcal{B}(Lambda_{c}^{+} rightarrow p K^{-} pi^{+})$ = $0.120 pm 0.006 pm 0.006$, $mathcal{B}(Lambda_{c}^{+} rightarrow Lambda(1670) pi^{+}) times mathcal{B}(Lambda(1670) rightarrow eta Lambda)/mathcal{B}(Lambda_{c}^{+} rightarrow p K^{-} pi^{+})$ = $(5.54 pm 0.29 pm 0.73 ) times 10^{-2}$, and $mathcal{B}(Lambda_{c}^{+} rightarrow eta Sigma(1385)^{+})/mathcal{B}(Lambda_{c}^{+} rightarrow p K^{-} pi^{+})$ = $0.192 pm 0.006 pm 0.016$. The mass and width of the $Lambda(1670)$ are also precisely determined to be $1674.3 pm 0.8 pm 4.9~{rm MeV}/c^{2}$ and $36.1 pm 2.4 pm 4.8~{rm MeV}$, respectively, where the uncertainties are statistical and systematic, respectively.
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