An exact solution is possible for all hydrogen-like atoms, i.e., atoms consisting of a nucleus and 1 electron. This includes deuterium: The solution is similar to that for hydrogen, but you have to use another reduced mass.

Yes, As we know, the deuteron is a two nucleon system, i.e. one proton and one neutron. We can obtain the energy eigenvalues and corresponding eigenfunctions for this system, analytically. Please see "Introduction to nuclear Physics, H. Enge" book. Deuterium atom is also a two particle system with a nucleus having a reduced mass about the half mass of a nucleon and a electron.