  In this paper we prove the existence of minimal non uniquely ergodic flipped
IETs. In particular, we build explicitly minimal non uniquely ergodic
$(10,k)$-IETs for any $1\leq k \leq 10$. This answers an open question posed in
[C.~Gutierrez, S.~Lloyd, V.~Medvedev, B.~Pires, and E.~Zhuzhoma.Transitive
circle exchange transformations with flips. Discrete Contin. Dyn. Syst.,26
(1):251--263, 2010]. As a consequence, we also derive the existence of
transitive non uniquely ergodic $(n,k)$-IETs, for any $n\geq 10$ and $1\leq
k\leq n$ if $n$ is even, and $1\leq k\leq n-1$ if $n$ is odd.
