  We prove the existence of an elliptic Reeb orbit for some contact forms on
the real projective three space $\mathbb{R} P^3$. The main ingredient of the
proof is the existence of a distinguished pseudoholomorphic curve in the
symplectization given by the $U$ map on ECH. Also, we check that the first
value on the ECH spectrum coincides with the smallest action of null-homologous
orbit sets for $1/4$-pinched Riemannian metrics and compute the ECH spectrum
for the irrational Katok metric example.
