'does not solely exist' is correct and usable in written English.
You can use it to express the idea that something has more than one cause or origin. For example, "Success does not solely exist because of hard work; it also depends on luck and timing."
And this political climate didn't solely exist on a national level, as famously exemplified by the conflict between Roger Griswold and Matthew Lyon.
These pins may or may not solely exist to inspire those feelings.
But justice does not exist solely for those with a particular approved outlook.
The Israeli people does not exist solely in the suffocating space between settlements and holy places, Sephardim and Ashkenazim, secular and religious".
Luckily for me, and for all of humanity, understanding does not exist solely along one plain of consciousness and learning, or more specifically growth, does not always take place in an instant.
So corruption does not exist solely in the hallowed halls of Congress, where lawmakers spend much of their time on the phone begging for money from plutocrats, but also in the very institution intended to protect the little guy, whether that means investors with small portfolios or families relying on pension funds to survive after their breadwinners' retirements.
Yes, it's true that Israel does not exist solely because of persecution and the Holocaust, so to say that "the aspiration for a Jewish homeland is rooted in a tragic history" might reinforce the common propaganda that Israel is a European colony, created without preamble in 1948.
In at least one district of Tanzania today, 80 per cent of the children who die before the age of five die at home without ever going to a hospital.23 But poverty does not exist solely in the developing world.
To the authors' knowledge, an approach does not yet exist that solely addresses a single tree species.
