  We perform a systematic analysis on supersymmetric states in N=4
supersymmetric Yang-Mills theory (SYM) on R x S^3. We find a new set of 1/16
BPS equations and determine the precise configuration of the supersymmetric
states by solving all 1/16 BPS equations when they are valued in Cartan
subalgebra of a gauge group and the fermionic fields vanish. We also determine
the number of supersymmetries preserved by the supersymmetric states varying
the parameters of the BPS solutions. As a byproduct we present the complete set
of such supersymmetric states in N=8 SYM on R x S^2 by carrying out dimensional
reduction.
