Adjective for "a property or a responsiblity that is not owned/assumed by anyone"
If there is no general word covering both meanings, separate words are also fine.
Example:
One of the obligations of Shia clerics is to manage ... properties or responsibilities.
I have thought of "abandoned" but that doesn't precisely mean what I wrote in the question title.
Postscript:
Thanks for your contributions! For more elaboration, this is related to a concept in Shia Jurisprudence called "hisba" which refers to obligations and responsibilities that have not been assumed by or are not naturally designated to any specific individual (like taking care of an orphan, an abandoned property, protecting ecosystem, etc) which then have to be taken care of by the clerics. I' looking for a word that can cover all these examples. I am thinking of adjectives in the vein of "stewardless," "ownerless," "un-owned," and "custody-less" etc but one that is probably more common and sound. And oh! I could just think of "unmanaged" or you may suggest a better one! :)
Solution 1:
The second definition of commons in the Oxford Dictionary On line is
Land or resources belonging to or affecting the whole of a community
This seems to be roughly what you are thinking about, at least in terms of property.
The same dictionary has a definition of Common Property as
A thing or things held jointly: for example ‘the atmosphere is the common property of every nation on earth’
The Encyclopaedia Britannica also has this definition of the common good
That which benefits society as a whole, in contrast to the private good of individuals and sections of society.
In your example sentence you could use the term communal for which is more difficult to find simple online references (partly, perhaps, because modern, western society places more emphasis on the individual than the community leading to the Tragedy of the Commons becoming an evermore widespread problem).
Solution 2:
In the UK it would be known as unclaimed property, and there is a body of law determining how it can be claimed or otherwise dealt with.
As regards responsibility I would suggest it would be an unassumed responsibility.
Solution 3:
Forsaken can be used for both. It is handy because it can be expanded for specificity. "A land forsaken of game". "A people forsaken of a nation's protection." I like it here because it conveys as sense of irregular circumstances. Property that would ordinarily be owned, or a responsibility that would normally be taken up, is in this case forsaken. It also tends to convey societal or ethical concern as opposed to merely physical abandonment.
Abandon, desert, forsake mean to leave without intending to return. Abandon suggests that the thing or person left may be helpless without protection . Desert implies that the object left may be weakened but not destroyed by one's absence . Forsake suggests an action more likely to bring impoverishment or bereavement to that which is forsaken than its exposure to physical dangers .
"Forsake." Merriam-Webster.com. Merriam-Webster, n.d. Web. 10 Dec. 2016.