TITLE: Bivariate polynomials with special properties
QUESTION [7 upvotes]: I recently came across some polynomials with some remarkable properties.
A polynomial  $P(u,v) \in \mathbb{R}[u,v]$ in 2 variables is remarkable if
the set of solutions to the system  $P(u,v)=P(v,u)=0$ is a finite number of points in $\mathbb{C}^2$ such that each point is of the form $(x, \overline{x}).$ 
Here are some examples of such polynomials:
$v^2-u$
$-2 u v+v^3+1$
$u^2-3 u v^2+v^4+2 v$
$3 u^2 v-4 u v^3-2 u+v^5+3 v^2$
$-u^3+6 u^2 v^2-5 u v^4-6 u v+v^6+4 v^3+1$
$-4 u^3 v+10 u^2 v^3+3 u^2-6 u v^5-12 u v^2+v^7+5 v^4+3 v$
$u^4-10 u^3 v^2+15 u^2 v^4+12 u^2 v-7 u v^6-20 u v^3-3 u+v^8+6 v^5+6 v^2$
A quite complicated algorithm is behind this sequence, any help identifying 
a formula for these polynomials would be helpful. 
Also, a proof that these polynomials are remarkable would be nice,
these are only checked by numeric computations in Mathematica.
Question: Can one classify the set of remarkable polynomials?

REPLY [3 votes]: Your  sequence   $p_n (u,v)$ can be defined by  $p_n (u,v) = vp_{n - 1} (u,v) - up_{n - 2} (u,v) + p_{n - 3} (u,v)$ with initial values $p_0 (u,v) = 1,p_1 (u,v) = v,p_2 (u,v) = v^2  - u.$
In my above remark I have overlooked a term in the fifth polynomial. This is now the same as the formula given by ARupinski.
Added later:
Extend the sequence $p_n (u,v)$  to negative indices by $p_{ - 1} (u,v) = p_{ - 2} (u,v) = 0$ and $p_{ - n} (u,v) = p_{ n - 3} (v,u)$ for $n >2 .$
Define a new sequence of polynomials $r_n (u,v)$ by the same recurrence and initial values $r_{ - 1} (u,v) = 1,r_0 (u,v) = 0,r_1 (u,v) =  - u.$  Extend it to negative values by  $r_{ - n} (u,v) = r_{n - 2} (v,u).$
Let $A$ be the matrix with rows  $(0,1,0),(0,0,1),(1, - u,v).$
Then $A^n$  is the matrix with the following rows: $\left( {p_{n - 3 + j} (u,v),r_{n - 2 + j} (u,v),p_{n - 2 + j} (u,v} \right)$ for  $0 \le j \le 2.$
It seems that the sequence $r_n (u,v)$ or the sequence $r_{2n} (u,v)/(1 - uv)$ has analogous properties with respect to the zeroes. Is it also related to the group representation?