I'm gonna go ahead and guess that the problem of determining all integer solutions to
[tex]x^3+y^3+z^3+t^3=1999[/tex]is open.
I have two reasons to suspect this: 1) These kinds of problems are generally
really, really, really difficult to solve (cf. FLT, ErdosStrauss, ...). 2) Let's suppose we were able to classify all integer solutions. We would then, in particular, be able to classify integer solutions in which t=1, or equivalently, we'd be able to solve
[tex]x^3+y^3+z^3=1998=3^3\cdot 74[/tex]in the integers. If we could solve that, then we'd know a thing or two about the rational solutions to
[tex]X^3 + Y^3 + Z^3 = 74.[/tex] I believe nobody knows if this last equation has any integer solutions. Indeed, in Cohen,
Number Theory: Analytic and Modern Tools (Springer, 2007), one finds the following piece of info:
As of 2007 the only integers n such that [itex]0 \leq n \leq 100[/itex] and [itex]n \not\equiv \pm4 \, (\text{mod } 9)[/itex] that are not known to be a sum of three cubes are n=33, 42, 52, and 74.

This leads to me suspect that people don't know much about the rational solutions of this equation either (but I could be wrong).