Solutions for $ a^b + b^a = a^c +b^c $ in $\mathbb{N}$

Question

Using python I found two solutions: $(4,16,8)$ and $(3,27,9)$. Also $a \neq b \neq c$ and $a,b,c \neq 1$. Are there any more solutions and if not, how to prove it?

Also does a similar equation $a^b + b^a = c^a + c^b$ has any solutions? I couldn't find any for $a,b,c < 100$