Context Free Grammer and PDA
The grammar $S \to aSa \mid bSb \mid cSc \mid a \mid b \mid c \mid \epsilon$ would generate (normal) palindromes.

A simple way to extend this to generate palindromes with non-repeating would be to add for each letter $x$ a new non-terminal $S_{not\text{-}x}$, and adapt the grammar: 

$S \to aS_{not\text{-}a}a \mid \dots$ 

$S_{not\text{-}a} \to bS_{not\text{-}b}b \mid \dots$

There are probably more efficient ways.