Geodesic
Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality
Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 566-580.

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We single out the class of so-called quasiregular Lagrangians, which have singularities on the zero section of the cotangent bundle to the manifold on which extremal networks are considered. A criterion for a network to be extremal is proved for such Lagrangians: the Euler–Lagrange equations must be satisfied on each edge, and some matching conditions must be valid at the vertices. 
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A. O. Ivanov; A. A. Tuzhilin; Lê Hông Vân. Nontrivial Critical Networks. Singularities of Lagrangians and a Criterion for Criticality. Matematičeskie zametki, Tome 69 (2001) no. 4, pp. 566-580. http://geodesic.mathdoc.fr/item/MZM_2001_69_4_a4/

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[2] Dubrovin B. A., Novikov S. P., Fomenko A. T., Sovremennaya geometriya, Nauka, M., 1986

[3] Ivanov A. O., Tuzhilin A. A., “Geometriya minimalnykh setei i odnomernaya problema Plato”, UMN, 47:2 (284) (1992), 53–115 | MR | Zbl

[4] Ivanov A. O., Tuzhilin A. A., Minimal Networks. The Steiner Problem and Its Generalizations, CRC Press, N.W., Boca Raton, Florida, 1994 | Zbl