Now suppose someone tells me that my solution "z=z0" is in fact accurate, so that my solution is correct and that there is no uncertainty in this third variable. Then the 3D covariance matrix will reduce to a 2D covariance matrix, for which the elements can be calculated by intersecting the 1 sigma ellipsoid with the plane specified by z=z0. I am not sure this is correct. Can someone confirm/correct?

Now suppose we generalize the problem further and that someone tells me the solution is z=z0 +/- delta z for a small z. This give me extra information. Does anyone know what is the best way to update my solution and covariance matrix?

Another way of formulating the problem: How to intersect a 3D multivariate normal distribution with a 1D one to get the 2D pdf?

Any help or pointers very much appreciated!