Cubic algebra and averaged Hamiltonian for the resonance 3: (-1) Penning-ioffe trap

For the 3: (-1) resonance Penning trap, we describe the algebra of symmetries which turns out to be a non-Lie algebra with cubic commutation relations. The irreducible representations and coherent states of this algebra are constructed explicitly. The perturbing inhomogeneous magnetic field of Ioffe type, after double quantum averaging, generates an effective Hamiltonian of the trap. In the irreducible representation, this Hamiltonian becomes a second-order ordinary differential operator of the Heun type.