Foliations induced by Vector Fields without Singularities

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A well-known type of foliations is the one that is induced by vector fields without singularities. However, I have already read this type of foliations from Geometric Theory of Foliations, Page 28. Unfortunately, I couldn't understand it very well. Therefore, I am actually looking for highly recommended references for the foliations induced by vector fields without singularities. Thanks in advance.

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One can think of foliations induced by vector fields as differential equations on manifolds (strictly speaking solutions to differential equations come with an extra natural group structure, whereas leaves of foliations have no canonical parameterization). For instance focusing on orbit equivalence instead of conjugacy as far as flows are concerned is in accordance with this perspective. In this line of thought here are two humble recommendations:

  • Hirsch, Smale & Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos,
  • Palis & de Melo's Geometric Theory Of Dynamical Systems.