= Solution
The scalar and vector potentials satisfy
$$
\mathbf E=-\nabla\phi-\partial_t\mathbf A,
\qquad
\mathbf B=\nabla\times\mathbf A.
$$
The canonical momentum and Hamiltonian are
$$
\mathbf p=m\dot{\mathbf r}+q\mathbf A,
\qquad
\boxed{H=\frac{|\mathbf p-q\mathbf A|^2}{2m}+q\phi}.
$$
For the uniform fields choose
$$
\mathbf A=(0,Bx,0),\qquad \phi=-Ex,
$$
which produces the stated Hamiltonian. Since it is time independent and $y,z$ are cyclic coordinates, three independent conserved quantities are
$$
\boxed{H,quad p_y,quad p_z}.
$$
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