What is the Oketz: The Oketz is the stem from which the Esrog grows from on the tree. If the Oketz was removed from the Esrog in a way that none of it remained on the Esrog, hence creating a grooved area, the Esrog is invalid. [Furthermore even if only part of the Oketz fell off and revealed part of the groove it is invalid.] However if the stem has been cut in such a way that an entire sliver of it remains and the groove of the Esrog is completely covered by this sliver then the Esrog is valid. However there are opinions which rule the Esrog is valid even if the entire Oketz was removed, as the Oketz is not part of the Esrog and hence cannot invalidate it due to a missing piece. Practically although we do not rule like this lenient opinion nevertheless if no other Esrog is available one may rely on this opinion and use it. In such a case one is allowed to use it with a blessing.
On Chol Hamoed: On Chol Hamoed one may use an esrog even if its stem has completely fallen off.
Second day in Diaspora: On the second day of Sukkos in the Diaspora, an Esrog which was invalidated due to a fallen Oketz may be used if there is no other Esrog available. In such a case it may used with a Bracha.
 The reason for this is because the Esrog is Chasar, is missing a piece. [ibid]
 As the actual Esrog is not missing any pieces. [ibid]
 The reason one may say a blessing is because some people are accustomed to say a blessing on all invalid species as will be explained. [ibid]
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