  We address the problem of characterisation of null-forms of conic
$3$-dimensional systems, that is, control-affine systems whose field of
admissible velocities forms a conic (without parameters) in the tangent space.
Those systems have been previously identified as the simplest control systems
under a conic nonholonomic constraint or as systems of zero curvature. In this
work, we propose a direct characterisation of null-forms of conic systems among
all control-affine systems by studying the Lie algebra of infinitesimal
symmetries. Namely, we show that the Lie algebra of infinitesimal symmetries
characterises uniquely null-forms of conic systems.
