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Equigeodesics on generalized flag manifolds with $G_2$-type $t$-roots

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.