  We study right-invariant (resp., left-invariant) Poisson-Nijenhuis structures
on a Lie group $G$ and introduce their infinitesimal counterpart, the so-called
r-n structures on the corresponding Lie algebra $\mathfrak g$. We show that
$r$-$n$ structures can be used to find compatible solutions of the classical
Yang-Baxter equation. Conversely, two compatible r-matrices from which one is
invertible determine an $r$-$n$ structure.
  We classify, up to a natural equivalence, all $r$-matrices and all $r$-$n$
structures with invertible $r$ on four-dimensional symplectic real Lie
algebras. The result is applied to show that a number of dynamical systems
which can be constructed by $r$-matrices on a phase space whose symmetry group
is Lie group $G$, can be specifically determined.
