Cylindrical type integrable classical systems in a magnetic field

We present all second order classical integrable systems of the cylindrical type in a three dimensional Euclidean space $\mathbb{E}_3$ with a nontrivial magnetic field. The Hamiltonian and integrals of motion have the form $H =\frac{1}{2}\left(\vec{p}+\vec{A}(\vec{x})\right)^2+W(\vec{x})$, $X_1=(p_\phi^A)^2+s_1^r(r, \phi, Z)p_r^A+s_1^\phi(r, \phi, Z)p_\phi^A+s_1^Z(r, \phi, Z)p_Z^A+m_1(r,\phi,Z)$, $X_2=(p_Z^A)^2+s_2^r(r, \phi, Z)p_r^A+s_2^\phi(r, \phi, Z)p_\phi^A+s_2^Z(r, \phi, Z)p_Z^A+m_2(r,\phi,Z)$. Infinite families of such systems are found, in general depending on arbitrary functions or parameters. This leaves open the possibility of finding superintegrable systems among the integrable ones (i.e. systems with 1 or 2 additional independent integrals).