Equivalence of existence of solution in rationals and solution in integers of polynomial equations

Question

Example -- We know that $\forall$ n $\in\mathbb{N}$, if solution of equations x$^{2}$ = n or x$^{2}$ + y$^{2}$ = n exist in rationals, then solution also exists in integers.

Question -- I was wondering what other equations have similar properties that existence of solutions in rationals implies existence of solution in integers.