  We report on a study of the isospin-violating and conserving decays of the
J/psi and psi(2S) charmonium state to Lambda Lambdabar pi0 and Lambda Lambdabar
eta, respectively. The data are based on 225 million J/psi and 106 million
psi(2S) events that were collected with the BESIII detector. The most accurate
measurement of the branching fraction of the isospin-violating process J/psi
->Lambda Lambdabar pi0 is obtained, and the isospin-conserving processes J/psi
->Lambda Lambdabar eta and psi(2S) ->Lambda Lambdabar eta are observed for the
first time. The branching fractions are measured to be B(J/psi ->Lambda
Lambdabar pi0) = (3.78+-0.27stat)+-0.29sys)10^-5; B(J/psi ->Lambda Lambdabar
eta) = (15.7+-0.79stat+-1.52sys)10^-5 and B(psi(2S) ->Lambda Lambdabar eta)=
(2.47+-0.34stat+-0.19sys)10^-5. No significant signal events are observed for
psi(2S) ->Lambda Lambdabar pi0 decay resulting in an upper limit of the
branching fraction of B(psi(2S) ->Lambda Lambdabar pi0) < 0.29X10^-5 at the 90%
confidence level. The two-body decay of J/psi-> Sigma(1385)^0 Lambdabar + c.c.
is searched for, and the upper limit is B(J/psi-> Sigma(1385)^0 Lambdabar +
c.c.) < 0.81X 10^-5 at the 90% confidence level.
