Magnetic field lines in a dynamical system?

The question is more about setting up the dynamical system than any of the Physics behind magnetic fields and stuff.

I get $\mathbf{B} = (-x^2 + y^2 + 1, 2xy, 0)^T$ and so $\frac{d\mathbf{x}}{ds} \times \mathbf{B} = (0,0, B_yx_s - B_xy_s)^T$ where $B_x,B_y$ are the x,y components of $\mathbf{B}$, and $x_s = \frac{dx}{ds}$.

Obviously from there I get that $B_yx_s = B_xy_s$. But from there I can't find anything that implies that $x_s = B_x, \; y_s = B_y$.