Are Transitions in a Hydrogen Atom Unique

I wrote a little Python program to find them:

#!/usr/bin/env python3

max_n = 200
for n in range(1, max_n):
    for m in range(n+1, max_n):
        for p in range(1, n):
            for q in range(p+1, m):
                if m*m*p*p*q*q +n*n*m*m*p*p == n*n*p*p*q*q + n*n*m*m*q*q:
                    print(f"Coincident spectral line for m={m}, n={n}, p={p}, q={q}")

Definitely not unique!

Coincident spectral line for m=90, n=6, p=5, q=9

Coincident spectral line for m=35, n=7, p=5, q=7

Coincident spectral line for m=56, n=8, p=7, q=14

Coincident spectral line for m=72, n=8, p=6, q=9

Coincident spectral line for m=72, n=9, p=6, q=8

Above equation shown below:

$m^2 p^2 q^2 - n^2 p^2 q^2 - n^2 m^2 q^2 + n^2 m^2 p^2=0$

Above is equivalent to:

$ (m^2 p^2 q^2) +(n^2 m^2 p^2) = (n^2 p^2 q^2) + (n^2 m^2 q^2)$

Or

$(mpq)^2+(nmp)^2=(npq)^2+(nmq)^2$ -------(A)

The above can be replaced by:

$(ac+bd)^2+(ad-bc)^2=(ac-bd)^2+(ad+bc)^2$ -------(B)

For equation (A) solution given by (User 14717) is (m,n,p,q)=(90,6,5,9)

Hence the solution for equation (B) is: (a,b,c,d)= (2,1,1080,1890)

But equation (B) has the condition:

$c^2(12a^2+b^2)=d^2(a^2+12b^2)$ -------(C)

Hence any new numerical solution to equation (C) will give new solution to Equation (A)