  All hypersurface homogeneous locally rotationally symmetric spacetimes which
admit conformal symmetries are determined and the symmetry vectors are given
explicitly. It is shown that these spacetimes must be considered in two sets.
One set containing Ellis Class II and the other containing Ellis Class I, III
LRS spacetimes. The determination of the conformal algebra in the first set is
achieved by systematizing and completing results on the determination of CKVs
in 2+2 decomposable spacetimes. In the second set new methods are developed.
The results are applied to obtain the classification of the conformal algebra
of all static LRS spacetimes in terms of geometrical variables. Furthermore all
perfect fluid nontilted LRS spacetimes which admit proper conformal symmetries
are determined and the physical properties some of them are discussed.
