Euclidean Killing vector field ($K(x)=A+Bx,B^T=-B$)

Every Killing vector field on Euclidean space is the sum of a constant translation and a constant infinitesimal rotation. The second covariant derivative of a Killing vector vanishes in flat space, so its components are affine; the Killing equation makes their linear coefficient an antisymmetric matrix.

An infinitesimal rotation in one coordinate plane plus a constant translation has integral curves of a vector field that rotate about a shifted centre in that plane and move uniformly in the remaining directions. Nonzero transverse drift gives helices; zero drift gives circles, with zero radius allowing straight or stationary curves.