  The aim of the present article is to construct quadratically integrable three
dimensional systems in non-vanishing magnetic fields which possess so-called
non-subgroup type integrals. The presence of such integrals means that the
system possesses a pair of integrals of motion in involution which are (at
most) quadratic in momenta and whose leading order terms, that are necessarily
elements of the enveloping algebra of the Euclidean algebra, are not quadratic
Casimir operators of a chain of its subalgebras. By imposing in addition that
one of the integrals has the leading order term $L_z^2$ we can consider three
such commuting pairs: circular parabolic, oblate spheroidal and prolate
spheroidal. We find all possible integrable systems possessing such structure
of commuting integrals and describe their Hamiltonians and their integrals.
  We show that our assumptions imply the existence of a first order integral
$L_z $, i.e. rotational invariance, of all such systems. As a consequence, the
Hamilton--Jacobi equation of each of these systems with magnetic field
separates in the corresponding coordinate system, as it is known to be the case
for all quadratically integrable systems without magnetic field, and in
contrast with the subgroup type, i.e. Cartesian, spherical and cylindrical,
cases, with magnetic fields.
  We also look for superintegrable systems within the circular parabolic
integrable class. Assuming the additional integral to be first order we
demonstrate that only previously known systems exist. However, for a particular
second order ansatz for the sought integral ($L^2+\ldots$) we find a minimally
quadratically superintegrable system. It is not quadratically maximally
superintegrable but appears to possess bounded closed trajectories, hinting at
hypothetical higher order superintegrability.
