Kalman filter with elliptical (quadratic) constraint

I am struggling with a problem in which the states and measurements
are both implicit in a constraint of the form

(mx-bx)^2/(1+sx)^2 + (my-by)^2/(1+sy)^2 = 1

where mx, bx are measurements and bx, by sx, sy are states to be
estimated. The states are generally constant but occassionally exhibit
discontinuities, and it is these discontinuities which I would like to
track.

I have been treating the constraint equation as a
"pseudo-measurement".

I have tried a standard extended KF, an extended "Bayes" filter, a
Schmidt KF (estimating bx and by only) and several variations.
Everything I have tried has been unstable. The matrix H*Px*HT (H -
Jacobian of constraint, Px state covariance) is very ill-conditioned.

I have experimented with various a priori covariances and with both
constant and Markov process models (with varying correlation times)
for the states.

Does anyone have any suggestions as to how to proceed?