The <Legendre transform> cancels the term linear in <velocity> and gives
$$
\boxed{H=\frac{p_x^2+p_y^2+(p_z+ax^2)^2}{2m}.}
$$
It equals the mechanical <kinetic energy>. The <Hamiltonian> has no explicit <time> dependence, so $H$ is conserved. This is also a direct consequence of the magnetic <force> doing no work: $q\mathbf v\cdot(\mathbf v\times\mathbf B)=0$.

Solution

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S7 / 1 / d / Solution

 Solution (source code)