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

Elliptic Reeb orbit on some real projective three-spaces via ECH