Existence of connections making a bundle endomorphism parallel 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 $\nabla N=0$?
 A: 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) 
A: No. To be parallel under a connection, $N$ would have to have the same type at each point. In other words, for any two points, there must be a linear map identifying tangent spaces which identifies $N$ at those two tangent spaces. If $N$ vanishes somewhere, it would have to vanish everywhere, for example.
