Palindromes and linear grammars
Consider Post correspondence problem (PCP). If $\alpha_1 \dots \alpha_n$ and $\beta_1 \dots \beta_n$ are the words of PCP, then $S \to \alpha_iS\beta_i^r$ and $S \to \#$ where $\#$ is a symbol not in any of $\alpha_i, \beta_i$ and $s^r$ is the string $s$ reversed, gives a grammar $G$ such that if $G$ generates a palindrome, then the corresponding instance of PCP has a solution. PCP is RE though.