Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $NC$?

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $NC$? Is there a theory behind such classification? (resolved by Ronald de Haan).

Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $NC$? (Update: It is not known)

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $NC$? Is there a theory behind such classification? (resolved by Ronald de Haan).

Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $NC$?

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $NC$? Is there a theory behind such classification?

Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $NC$?

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $NC$? Is there a theory behind such classification? (resolved by Ronald de Haan).

Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version $P$-complete or in $NC$?

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in $P$ and provably $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in $P$ and provably in $NC$? Is there a theory behind such classification?

Integer programming is $NP$ complete however fixed parameter tractable in number of variables. Is the fixed parameter version in parametrized analogue of $P$-complete or in parametrized analogue of $NC$?

What are some examples of problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $P$ complete and problems that are $NP$ complete but whose fixed parameter version is in parametrized analogue of $P$ and provably in parametrized analogue of $NC$? Is there a theory behind such classification?