# Determination of a unique electric and magnetic field produced by a point charge

Let's suppose I have a point charge moving arbitrarily in space. As far I know, Maxwell's equations don't determine a unique electric and magnetic field, they only determine a family of electric and magnetic fields. What other information do I need in order to know uniquely the electric and magnetic field produced by a moving (the particle is moving arbitrarily) point charge?

I'm not asking for a method for solving it, I just want to know what other information (apart from Maxwell equations) do I need.

If we know the position of the charged particle at any time, that is, if we know the value of the function $\mathbf r(t)$ for all $t$, one way to find the EM fields is based on knowledge of the fields at one time instant $t_0$. Let the field be such that
$$\mathbf E(\mathbf x, t_0) = \mathbf C_E(\mathbf x),~~\mathbf B(\mathbf x, t_0)= \mathbf C_B(\mathbf x)~~\forall \mathbf x.$$
With knowledge of the functions $\mathbf C_E(\mathbf x), \mathbf C_B(\mathbf x)$, using the temporal Maxwell equations
$$\nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t}$$ $$\nabla \times \mathbf B = \mu_0 \mathbf j + \mu_0 \epsilon_0 \frac{\partial \mathbf E}{\partial t}$$ and $$\mathbf j(\mathbf x,t) = q\mathbf v(t)\delta(\mathbf x-\mathbf r(t))$$ we can find $\mathbf E(\mathbf x, t), \mathbf B(\mathbf x, t)$ for any $t$.