  We study homogeneous curves in generalized flag manifolds $G/K$ with
$G_2$-type $t$-roots, which are geodesics with respect to each $G$-invariant
metric on $G/K$. These curves are called equigeodesics. The tangent space of
such flag manifolds splits into six isotropy summands, which are in one-to-one
correspondence with $t$-roots. Also, these spaces are a generalization of the
exceptional full flag manifold $G_2/T$. We give a characterization for
structural equigeodesics for flag manifolds with $G_2$-type $t$-roots, and we
give for each such flag manifold, a list of subspaces in which the vectors are
structural equigeodesic vectors.
