A $ d \times d $ non-conservative and non strictly hyperbolic but hyperbolic diagonal system appearing in the theory of dislocations is studied in the framework of fractional $BV$ spaces. A definition of entropy solution is proposed for this non-conservative system. Existence of entropy solutions in $BV^s$ for all $ 0 2$. The infinite family of entropies for $d=2$ is not enough to ensure uniqueness due to the loss of stric hyperbolicity. Entropy Riemann solvers that recover main properties of vanishing viscosity solutions exhibit more waves than usual, already for $d=2$.