The main reference that we will follow will be Mackenzie's book:
Kirill C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series,
Cambridge University Press (2013), doi: 10.1017/CBO9781107325883
Other references include
Marius Crainic and Rui L. Fernandes, Lectures on Integrability of Lie brackets (2006), arXiv: math/0611259v1 [math.DG]
Charles-Michel Marle, Calculus on Lie algebroids, Lie groupoids and Poisson manifolds, Dissertationes Mathematicae457 (2008) 1--57, doi: 10.4064/dm457-0-1
Ieke Moerdijk and Janez Mrčun, Introduction to Foliations and Lie Groupoids, Cambridge University Press (2003), doi: 10.1017/CBO9780511615450
Manuel de León, Juan C. Marrero and Eduardo Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, J. Phys. A: Math. Gen.38 (2005) R241, doi: 10.1088/0305-4470/38/24/R01
Katarzyna Grabowska, Paweł Urbański and Janusz Grabowski, Geometrical mechanics on Lie algebroids, Int. J. Geom. Meth. Mod. Phys.3 (2006) 559--575, doi: 10.1142/S0219887806001259
Jorge Cortés, Manuel de León, Juan C. Marrero, D. Martín de Diego, and Eduardo Martínez, A survey of Lagrangian mechanics and control on Lie algebroids and groupoids, Int. J. Geom. Methods Mod. Phys.3, no. 3 (2006), 509--558, doi: 10.1142/S0219887806001211
Juan C. Marrero, David Martín de Diego and Eduardo Martínez, Discrete Lagrangian and Hamiltonian mechanics on Lie groupoids, Nonlinearity19 (2006) 1313, doi: 10.1088/0951-7715/19/6/006
María A. Salazar, Pfaffian groupoids, Ph.D. Thesis, Utrecht University, 2013, arXiv: 1306.1164 [math.DG]
Marius Crainic and Oren Yudilevich, Lie Pseudogroups à la Cartan, Mem. Amer. Math. Soc.303 (2024), doi: 10.1090/memo/1527
Juan Núñez Valdés, Ángel F. Tenorio Villalón, and José Antonio Vilches Alarcón, Elementos de la teoría de grupoides y algebroides. Servicio de Publicaciones de la Universidad de Cádiz (2006). URI: http://hdl.handle.net/10498/26397