Geodesic
Flows on compact solvmanifolds
Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 549-556

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The author proves that any $G$-induced flow on a compact solvable homogeneous space $G/D$ is ergodic on a submanifold $P(x)\subset G/D$. For almost any flow the closure of the orbit of the fixed point $\exp(tx)D$ is the submanifold $P(x)\subset G/D$. Bibliography: 10 titles. 
@article{SM_1985_51_2_a13,
     author = {A. N. Starkov},
     title = {Flows on compact solvmanifolds},
     journal = {Sbornik. Mathematics},
     pages = {549--556},
     publisher = {mathdoc},
     volume = {51},
     number = {2},
     year = {1985},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1985_51_2_a13/}
}
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A. N. Starkov. Flows on compact solvmanifolds. Sbornik. Mathematics, Tome 51 (1985) no. 2, pp. 549-556. http://geodesic.mathdoc.fr/item/SM_1985_51_2_a13/