Assume that magnetic monopoles exist. Let us have one charge(for example, an electron) of electric charge $q_e$ at the origin and one magnetic monopole particle of magnetic charge $q_m$ separated by a vector $\vec{a}$.
Then the electric field generated by the charge will be of the form : $\vec{E}(\vec{r})=\frac{q_e}{4\pi\epsilon_0}\frac{\vec{r}}{||\vec{r}||^3}$
and the magnetic field generated by the magnetic monopole : $\vec{B}(\vec{r})=\frac{q_m}{4\pi\mu_0}\frac{\vec{r}-\vec{a}}{||\vec{r}-\vec{a}||^3}$.
So by considering the standard definition of the Poynting vector $\vec{S} = \vec{E}\times\vec{B}$, we have $\vec{S}(\vec{r}) = -\frac{q_eq_m}{(4\pi)^2\epsilon_0\mu_0}\frac{\vec{r}\times\vec{a}}{||\vec{r}||^3||\vec{r}-\vec{a}||^3}$, which is non-zero for all vectors $\vec{r}$ non parallel to $\vec{a}$.
According to my understanding, this means that EM energy is radiated away, which cannot happen due to energy conservation. My question is: which of the following statement is correct and why :
(1) The poynting vector formula is modified when considering that magnetic monopoles exist.
(2) The magnetic field of a magnetic monopole is not radial as I assumed.
(3) Energy would indeed be radiated away, making the 2 particles disappear in EM radiation, similar to matter-antimatter interaction.
(4) Something else.