Let $\Delta=\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=-\frac{1}{2}(a+b+c)\left[(a-b)^2+(b-c)^2+(c-a)^2\right]$
(A) If $a+b+c \neq 0$ and $a^2+b^2+c^2=a b+b c+c a$
$$
\Rightarrow \quad \Delta=0 \text { and } a=b=c \neq 0
$$
The equations represents identical planes.
(B) $a+b+c=0$ and $a^2+b^2+c^2 \neq a b+b c+c a$
$$
\Rightarrow \quad \Delta=0
$$
The equations have infinitely many solutions.
$$
\begin{array}{rlrl}
& & a x+b y & =(a+b) z \\
\Rightarrow & b x+c y & =(b+c) z \\
\Rightarrow & & \left(b^2-a c\right) y & =\left(b^2-a c\right) z \Rightarrow y=z \\
\Rightarrow & a x+b y+c y & =0 \\
& a x & =a y \Rightarrow x=y=z
\end{array}
$$
(C) $a+b+c \neq 0$ and $a^2+b^2+c^2 \neq a b+b c+c a$
$$
\Rightarrow \quad \Delta \neq 0
$$
The equation represent planes meeting at only one point.
(D) $a+b+c=0$ and $a^2+b^2+c^2=a b+b c+c a$
$$
\Rightarrow \quad a=b=c=0
$$
The equations represent whole of the three dimensional space.