It is possible to define the Lie groupoid of a singular foliation and associates to it its Lie algebroid when it is smooth. This Lie algebroid satisfies the property 2.
https://projecteuclid.org/download/pdf_1/euclid.jdg/1090348356Debord - Holonomy Groupoids of Singular Foliations
http://users.uoa.gr/~iandroul/AS-holgpd-final.pdfAndroulidakis and Skandalis - The holonomy groupoid of a singular foliation
https://en.wikipedia.org/wiki/Lie_algebroid#Lie_algebroid_associated_to_a_Lie_groupoidLie algebroid associated to a Lie groupoid
It is a conjecture of Androulidakis and M. Zambon (see ( Lavau'sLavau's thesis p. 65) that not every singular foliation arises from a Lie algebroid. In the second reference, there is a condition which implies the smoothness of the holonomy groupoid.
https://tel.archives-ouvertes.fr/tel-01447963/documentLavau - Lie $\infty$-algébroïdes et feuilletages singuliers
https://www3.ubu.es/ifwgp2012/transparencias%20web/Zambon.pdf
