  We study homogeneous curves on some classes of reductive homogeneous spaces
G=H which are geodesics with respect to any G-invariant metric on G=H. These
curves are called equigeodesics. The spaces we consider are certain Stiefel
manifolds VkRn, generalized Wallach spaces and spheres. We give a
characterization for algebraic equigeodesics on V2Rn, V4R6, SO(6)= SO(3) ?
SO(2), W6 = U(3)= U(1)3, W12 = Sp(3)= Sp(1)3, S2n+1 ?= U(n + 1)= U(n) and S4n+3
?= Sp(n + 1)= Sp(n).
