  Two theorems witnessing the abundance of geometrically trivial strongly
minimal autonomous differential equations of arbitrary order are shown. The
first one states that a generic algebraic vector field of degree $d\geq 2$ on
the affine space of dimension $n \geq 2$ is strongly minimal and geometrically
trivial. The second one states that if $X_0$ is the complement of a smooth
hyperplane section $H_X$ of a smooth projective variety $X$ of dimension $n
\geq 2$ then for $d \gg 0$, the system of differential equations associated
with a generic vector field on $X_0$ with poles of order at most $d$ along
$H_X$ is also strongly minimal and geometrically trivial.
