Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?



Existence of connections making a bundle endomorphism parallel
My answer is almost contained in the answers/comments of Mariano Suárez-Alvarez, Ben McKay and Robert Bryant; I summarize the answers and give a reference. 


There exists a torsion-free connection such that a given endomorphism $A$ is parallel if and only if the following two conditions are fulfilled:
(1) The Jordan type of the endomorphism is the same at all points
(2) The Nijenhujs torsion of the endomorphism vanishes. 
(3) For every k and every eigenvalue $c$ the distribution $Kern(A-c Id)^k $ is integrable. 


A reference is Thompson, Gerard The integrability of a field of endomorphisms. Math. Bohem. 127 (2002), no. 4, 605–611 and it is downloadable from <https://eudml.org/doc/249030> (actually, this result was reproved and republished many times, first time possible by Shirokov in the 50th who unfortunately published most his works in obscure places and possibly the last time by Boubel arXiv:1003.0979) 


