15-10E

Modified Penning trap with a spatially uniform magnetic field $\mathbf B$ inclined  with respect to the axis of rotational symmetry of the electrodes is considered. The inclination angle can be arbitrary. Canonical transformation of phase variables transforming Hamiltonian of considered system into a sum of three uncoupled  harmonic oscillators is found.  We determine  the region of  stability in space of two parameters controlling the dynamics: the trapping parameter $\kappa$ and the squared sine of the inclination angle $\vartheta_0$. If  the angle $\vartheta_0$  is smaller than $54$ degrees, a charge occupies finite spatial volume within processing chamber. A rigid hierarchy of trapping frequencies is broken if $\mathbf B$ is inclined at the critical angle: the magnetron frequency reaches the corrected cyclotron frequency while the axial frequency exceeds them. Apart from this resonance we reveal the family of resonant curves in the region of stability.
In the relativistic regime the system is not linear. We show that it is not integrable in the Liouville sense.  The averaging over the fast variable allows to reduce the system to two degrees of freedom. An analysis of the Poincar\'e cross-section of the averaged systems shows the regions of effective stability of the trap.