Geodesic
Many-Dimensional Poincar\'e Construction and Singularities of Lifted Fields For Implicit Differential Equations
Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 19 (2006), pp. 131-170

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The paper is devoted to singular points of the so-called lifted vector fields, which arise in studying systems of implicit differential equations by using the method of lifting the equation to a surface, a generalization of the construction used by Poincaré for a single implicit equation. The author study the phase portraits and renormal forms of such fields in a neighborhood of their singular points. In conclusion, the paper considers the lifted vectors fields generated by Euler–Lagrange and Euler–Poisson equations and fast-slow systems. 
@article{CMFD_2006_19_a5,
     author = {A. O. Remizov},
     title = {Many-Dimensional {Poincar\'e} {Construction} and {Singularities} of {Lifted} {Fields} {For} {Implicit} {Differential} {Equations}},
     journal = {Contemporary Mathematics. Fundamental Directions},
     pages = {131--170},
     publisher = {mathdoc},
     volume = {19},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CMFD_2006_19_a5/}
}
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A. O. Remizov. Many-Dimensional Poincar\'e Construction and Singularities of Lifted Fields For Implicit Differential Equations. Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 19 (2006), pp. 131-170. http://geodesic.mathdoc.fr/item/CMFD_2006_19_a5/