From a discretization of an incompressible Navier-Stokes equation I get the following matrix
$$M = \begin{pmatrix}A & B_1^\intercal \\ B_2 & C\end{pmatrix}$$
Because my density is independent of the pressure, $C = 0$. But $B_1^\intercal$ and $B_2$ are not transposed matrices, i. e., $B_1 \ne B_2$ and $A$ is non-symmetric, because we use upwinding for several physical properties like density or viscosity.
There are plenty of papers how to solve the symmetric saddle point problem. There are several on how to solve this for a non-symmetric $A$, like Bramble, Pasciak, Vassilev, 1999 and its improvements.
I could not find a source which allows $B_1 \ne B_2$. For engineering problems with a non-constant density this should be a common problem. What's the best way to solve this kind of system?