When we solve the Schrodinger equation for the Helium atom, we usually resort to approximation methods citing that it cannot be exactly solved analytically.
When it is said that it "cannot" be solved does that translate to "an analytical solution does not exist" or "nobody has yet found a way to solve it exactly analytically"?
Solving Helium atom means solving a three-body problem - moreover, quantum three body problem, which is unlikely to be easier than a classical one:
Unlike two-body problems, no general closed-form solution exists, as the resulting dynamical system is chaotic for most initial conditions, and numerical methods are generally required.
A quantum-mechanical analogue of the gravitational three-body problem in classical mechanics is the helium atom, in which a helium nucleus and two electrons interact according to the inverse-square Coulomb interaction. Like the gravitational three-body problem, the helium atom cannot be solved exactly.
This last passage contains reference to Griffiths' Introduction to Quantum Mechanics. The passage that I found in Griffiths' book, preceding equation (7.16) is:
It is curious that such a simple and important problem has no known exact solution 3. The trouble comes from the electron-electron repulsion, [...].
Footnote 3 reproduces what has been also mentioned in Wikipedia - that the problem is solvable for some special kinds of potential, but not the Coulomb one.
