  We prove a conjecture of Medvedev and Scanlon for endomorphisms of connected
commutative linear algebraic groups $G$ defined over an algebraically closed
field $\mathbb{k}$ of characteristic $0$. That is, if $\Phi\colon
G\longrightarrow G$ is a dominant endomorphism, we prove that one of the
following holds: either there exists a non-constant rational function $f\in
\mathbb{k}(G)$ preserved by $\Phi$ (i.e., $f\circ \Phi = f$), or there exists a
point $x\in G(\mathbb{k})$ whose $\Phi$-orbit is Zariski dense in $G$.
