The questions are:

1. Is there anything else special about $a^2b+b^2c+c^2a = 0$?
2. In what other contexts does it appear?

For question 1, Given the $q$-series of degree $9$, $$ \frac ac=a^3-b^3,\quad\frac ba=b^3-c^3,\quad\frac cb=c^3-a^3 $$ which leads to the telescoping sum $$ \frac ac+\frac ba+\frac cb = 0 $$ as noticed already. Another feature is multi-section. Let $$ A := -q^{-1/9}\frac{f(-q^{1/3})}{f(-q^3)} = a + b + c $$ where $a,b,c$ form a 3-section of the $q$-series $A$.

For question 2, assume $a^2b+b^2c+c^2a = 0.$ Now define $$ x := \sqrt[3]{a^2b},\quad y := \sqrt[3]{b^2c},\quad z := \sqrt[3]{c^2a}. $$ By definition, $ x^3 + y^3 + z^3 = 0 $ which is a cubic equation equivalent to the elliptic curve "27a3", the Fermat cubic. Something similar happens with the Klein quartic curve and the Fermat septic curve.

The questions are:

1. Is there anything else special about $a^2b+b^2c+c^2a = 0$?
2. In what other contexts does it appear?

For question 1, Given the $q$-series of degree $9$, $$ \frac ac=a^3-b^3,\quad\frac ba=b^3-c^3,\quad\frac cb=c^3-a^3 $$ which leads to the telescoping sum $$ \frac ac+\frac ba+\frac cb = 0 $$ as noticed already. Another feature is multi-section. Let $$ A := -q^{-1/9}\frac{f(-q^{1/3})}{f(-q^3)} = a + b + c $$ where $a,b,c$ form a 3-section of the $q$-series $A$.

For question 2, refer to L. E. Dickson, History of the Theory of Numbers, Volume II, Chapter XXI, section "Impossibility of $x^3+y^3=z^3$", pp. $545$-$550$. On page $546$ it states

$\quad$ J. A. Euler$^9$ noted that, if $\,p^3+q^3+r^3=0\,$ is possible, $\,x=p^2q,$ $y=q^2r,$ $z=r^2p\,$ satisfy $\,x/y+y/z+z/x=0\,$ or $\,x^2z+y^2x+z^2y=0.$

Something similar happens with the Klein quartic curve and the Fermat septic curve.

The questions are:

1. Is there anything else special about $a^2b+b^2c+c^2a = 0$?
2. In what other contexts does it appear?

For question 1, Given the $q$-series, $$ \frac ac=a^3-b^3,\quad\frac ba=b^3-c^3,\quad\frac cb=c^3-a^3 $$ which leads to the telescoping sum $$ \frac ac+\frac ba+\frac cb = 0 $$ as noticed already. Another feature is multi-section. Let $$ A := -q^{-1/9}\frac{f(-q^{1/3})}{f(-q^3)} = a + b + c $$ where $a,b,c$ form a 3-section of the $q$-series $A$.

For question 2, assume $a^2b+b^2c+c^2a = 0.$ Now define $$ x := \sqrt[3]{a^2b},\quad y := \sqrt[3]{b^2c},\quad z := \sqrt[3]{c^2a}. $$ By definition, $ x^3 + y^3 + z^3 = 0 $ which is a cubic equation equivalent to the elliptic curve "27a3", the Fermat cubic.

The questions are:

1. Is there anything else special about $a^2b+b^2c+c^2a = 0$?
2. In what other contexts does it appear?

For question 1, Given the $q$-series of degree $9$, $$ \frac ac=a^3-b^3,\quad\frac ba=b^3-c^3,\quad\frac cb=c^3-a^3 $$ which leads to the telescoping sum $$ \frac ac+\frac ba+\frac cb = 0 $$ as noticed already. Another feature is multi-section. Let $$ A := -q^{-1/9}\frac{f(-q^{1/3})}{f(-q^3)} = a + b + c $$ where $a,b,c$ form a 3-section of the $q$-series $A$.

For question 2, assume $a^2b+b^2c+c^2a = 0.$ Now define $$ x := \sqrt[3]{a^2b},\quad y := \sqrt[3]{b^2c},\quad z := \sqrt[3]{c^2a}. $$ By definition, $ x^3 + y^3 + z^3 = 0 $ which is a cubic equation equivalent to the elliptic curve "27a3", the Fermat cubic. Something similar happens with the Klein quartic curve and the Fermat septic curve.