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We introduce a representation of a 2D steady vector field \\( v\\) by two scalar fields \\(a\\), \\(b\\), such that the isolines of \\(a\\) correspond to stream lines of \\( v\\), and \\(b\\) increases with constant speed under integration of \\( v\\). This way, we get a direct encoding of stream lines, i.e., a numerical integration of \\( v\\) can be replaced by a local isoline extraction of \\(a\\). To guarantee a solution in every case, gradient-preserving cuts are introduced such that the scalar fields are allowed to be discontinuous in the values but continuous in the gradient. Along with a piecewise linear discretization and a proper placement of the cuts, the fields \\(a\\) and \\(b\\) can be computed. We show several evaluations on non-trivial vector fields.