But in addition to that now I'd like to compute a nonlinear static load case and export the linearized stiffness matrix around the static equilibrium point.
Hi Johannes, I'm not sure this is exactly what you want, but if you are just looking for the differential stiffness you can use the alter in the following article:
TYPE PARM,,I,N,PUNCHFLG=1 $ GENERATE DMIG PUNCHED OUTPUT
TYPE PARM,,I,N,SORTFLG=1 $ INPUT MATRICES ARE IN INTERNAL SORT
MATMOD KDGGXX,EQEXIN,USET,,,/,
/16/PUNCHFLG/SORTFLG//////////'KDDEXT'/'G' $
In SOL106, I think the different stiffnesses are kept in separate matrices (linear, differential, material), though at some point I'm sure they get combined.