  We prove an existence result for the Backus interior problem in the Euclidean
ball. The problem consists in determining a harmonic function in the ball from
the knowledge of the modulus of its gradient on the boundary. The problem is
severely nonlinear. From a physical point of view, the problem can be
interpreted as the determination of the velocity potential of an incompressible
and irrotational fluid inside the ball from measurements of the velocity
field's modulus on the boundary. The linearized problem is an irregular oblique
derivative problem, for which a phenomenon of loss of derivatives occurs. As a
consequence, a solution by linearization of the Backus problem becomes
problematic. Here, we linearize the problem around the vertical height solution
and show that the loss of derivatives does not occur for solutions which are
either (vertically) axially symmetric or oddly symmetric in the vertical
direction. A standard fixed point argument is then feasible, based on ad hoc
weighted estimates in H\"older spaces.
