  The Brio system is a two-by-two system of conservation laws arising as a
simplified model in ideal magnetohydrodynamics (MHD). The system has the form
\begin{align*} \partial_t u+\partial_x \Big({\textstyle
\frac{u^2+v^2}{2}}\Big)=0,\\ \partial_t v+\partial_x \big(v(u-1)\big)=0.
\end{align*} It was found in previous works that the standard theory of
hyperbolic conservation laws does not apply to this system since the
characteristic fields are not genuinely nonlinear on the set $v=0$. As a
consequence, certain Riemann problems have no weak solutions in the traditional
class of functions of bounded variation.
  It was argued in Nonlinearity 9, 1547--1563 (1996) that in order to solve the
system, singular solutions containing Dirac masses along the shock waves might
have to be used. Solutions of this type were exhibited in Proc. Edinb. Math.
Soc. 55, 711--729 (2012) and Russ. J. Math. Phys. 22, 518--527 (2015), but
uniqueness was not obtained.
  In the current work, we introduce a nonlinear change of variables which makes
it possible to solve the Riemann problem in the framework of the standard
theory of conservation laws. In addition, we develop a criterion which leads to
an admissibility condition for singular solutions of the original system, and
it can be shown that admissible solutions are unique in the framework developed
here.
