  In this paper we present the construction of all nonstandard integrable
systems in magnetic fields whose integrals have leading order structure
corresponding to the case (i) of Theorem 1 in [A Marchesiello and L \v{S}nobl
2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203]. We find that the
resulting systems can be written as one family with several parameters. For
certain limits of these parameters the system belongs to intersections with
already known standard systems separating in Cartesian and / or cylindrical
coordinates and the number of independent integrals of motion increases, thus
the system becomes minimally superintegrable. These results generalize the
particular example presented in section 3 of [A Marchesiello and L \v{S}nobl
2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203].
