  Four components of the axisymmetric Einstein equations in 2+1 dimensions with
negative cosmological constant can be written as $\nabla_aM=\dots$ and
$\nabla_aJ=\dots$, where the dots stand for stress-energy terms, and $M$ and
$J$ are scalars. In vacuum, they reduce to the constant mass and angular
momentum parameters of the BTZ solution of the same name. The integrability
conditions for the Einstein equations give rise to two conserved stress-energy
currents $\nabla_aj^a_{(M)}=0$ and $\nabla_aj^a_{(J)}=0$. The angular momentum
current is just the Noether current due to axisymmetry, but the mass current is
unexpected in the presence of rotation. The conserved quantity $M$ exists in
all dimensions in spherical symmetry, known as the Misner-Sharp, Hawking or
Kodama mass, but in 2+1 dimensions $M$ exists also in axisymmetry, even with
rotation. We use $M$ and $J$ to give a fully constrained formulation of the
axisymmetric Einstein equations in 2+1 dimensions, where the Einstein equations
are solved by explicit integration from the centre along time slices. We use
the two conserved matter currents in the construction of a high-resolution
shock-capturing formulation of the Einstein-perfect fluid system, in which $M$
and $J$ momentum are then exactly conserved by construction. We demonstrate
convergence of the code in the test cases of generic dispersion and collapse
and stable and unstable rotating stars.
