  This article is devoted to the generalization of results obtained in 2002 by
Jabin, Otto and Perthame. In their article they proved that planar vector
fields taking value into the unit sphere of the euclidean norm and satisfying a
given kinetic equation are locally Lipschitz. Here, we study the same question
replacing the unit sphere of the euclidean norm by the unit sphere of
\emph{any} norm. Under natural asumptions on the norm, namely smoothness and a
qualitative convexity property, that is to be of power type $p$, we prove that
planar vector fields taking value into the unit sphere of such a norm and
satisfying a certain kinetic equation are locally $\frac{1}{p-1}$-H\"older
continuous. Furthermore we completely describe the behaviour of such a vector
field around singular points as a \emph{vortex} associated to the norm. As our
kinetic equation implies for the vector field to be curl-free, this can be seen
as a selection principle for curl-free vector fields valued in spheres of
general norms which rules out line-like singularities.
