The phrase "a unique responsibility to" is correct and usable in written English.
You can use it when discussing a specific duty or obligation that is distinct or singular in nature.
Example: "As the project leader, I have a unique responsibility to ensure that all team members are heard and valued."
Alternatives: "a distinct obligation to" or "a special duty to".
"I have a unique responsibility to try to make this town work," said the president.
She added, "I do feel that, as a female athlete, I have a unique responsibility to the feminist movement.
What he's said: "I believe the United States has a unique responsibility to act — indeed, we have a moral obligation.
Mr. Blunt said the special master would have "a unique responsibility to arrive at a fair settlement from all perspectives".
It isn't simply that America, by virtue of symbolizing liberty, has a unique responsibility to shape the affairs of humankind.
But Duke said that Wal-Mart has a unique responsibility to address the problems of farming because it is the biggest grocer in the world.
After all, they argue, the military has the unique responsibility to train people to fight and die in war.
NATO's unique responsibility to act is obvious: Three of the five nuclear weapon states recognized by the Nuclear Nonproliferation Treaty and eight of the 14 states that have nuclear weapons on their territory are NATO members.
With the new lighting retrofit, "the owners of the Empire State Building have a unique responsibility" not to mess up, said Paul Gregory, a principal in Focus Lighting, an architectural lighting design firm based in New York.
The costs involved in designing, building, equipping, and ultimately decommissioning nuclear facilities place a unique responsibility on designers to extract the maximum use from these valuable assets: and in the longer term to simplify the decommissioning process as much as possible.
Health professionals and their organisations have a unique responsibility in trying to ensure that children receive treatment.
