Is this language regular?
Nice question. It is not regular.

Notice that this language consists of words where some nonempty prefix is an even palindrome.

Intersect $L$ with $(01)^{+} (10)^{+}$. If a word of this form has a palindromic prefix: $(01)^n (10)^m = u u^R v$ and $u \neq \varepsilon$, then the center of palindrome must be between the group of $01$ and the group of $10$. For example, take the word $0101011010101010$. The only possibility of breaking a prefix into a palindrome is $(010101\cdot 101010)\cdot1010$. (I leave the proof of this statement to you.)

Therefore, $L \cap (01)^{+} (10)^{+}=\{(01)^n (10)^m : m &gt; n \geq 1\}$. However, this language is not regular - for example, each prefix $(01)^n$ is in a different Myhill-Nerode class.