With $\rho=q\delta(r-r_a)$, $J=q\dot{\vec{r_a}}\delta(r-r_a)$ and $m\ddot{\vec{r_a}}=q(E+\dot{\vec{r_a}}\times B)$,
Lets say that the initial condition is $r_a(t=0)=0,v_a(t=0)=\vec{v_0}$
If your particle moves at a constant velocity, you can work in a reference frame in which it is at rest. The solution would be just a Coulomb field ($\vec{B}=0, \vec{E}=kq\hat{r}/r^2 $).
To obtain a solution for a moving charge, you can switch to a moving reference frame by Lorentz-transforming the fields.
I doubt that it is possible to to find an analytic solution for a completely arbitrary trajectory. The best thing one can do in that case is the Liénard–Wiechert potential but as far as I know it is given in terms of the so called retarded time and in general it is not possible to find an explicit expression for it. If we do allow a constant velocity the whole thing looks a bit different.
Basically a constant velocity would mean that we go into the rest frame of the particle (every charged particle I know of has a mass, so it moves slower than light). In the rest frame the particle does not move at all, it just stays where it is and you can just solve the Maxwell equations for a stationary point charge and then take a Lorentz transform back into the original frame where the particle is moving. One could also take the Liénard–Wiechert potential, plug in a constant velocity and in that special case there should be a possibility to find out what the retarded time is.
In concusion: You can read a little bit about the Liénard–Wiechert potential (maybe start with the Wikipedia article) which is a solution but given with respect to the retarded time, a parameter that is only given by an implicit relation which in general you can't solve. In the special case of constant velocity one even can solve it completely.
I hope that helps!
