This paper considers the problem of testing for nonzero values of the equicorrelation coefficients in the three-stage standard symmetric multivariate normal (SSMN) distributions and proposes locally most mean powerful (LMMP) and point optimal (PO) tests. It also demonstrates that under a special situation, the LMMP test is equivalent to SenGupta's (1988) locally best (LB) test for the case of two-stage SSMN distribution. An empirical power comparison of SenGupta's LB test with two versions of the PO test and the power envelope (PE) shows that the two PO tests are approximately uniformly the most powerful because this power curve is the closest to that of PE.