  We investigate metric properties of level sets of horizontally differentiable
maps defined on the first Heisenberg group $(\Bbb{H}^1,d_{cc})$ equipped with
the standard sub-Riemannian structure. In particular, we present an exhaustive
analysis in a new case of a map $F\in C^1_H(\Bbb{H}^1, \Bbb{R}^2)$ with
surjective horizontal differential (an analogue of the classical implicit
function theorem). Among other results, we show that a level set of such map is
locally a simple curve of Hausdorff sub-Riemannian dimension 2, but,
surprisingly, in general its two-dimensional Hausdorff measure can be zero or
infinity. Therefore, those level sets (called \textsf{vertical curves}) can be
of rough nature and not belong to the class of intrinsic regular manifolds.
