Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n ≡ b { mod}\, a$ if $(a,m)=1$

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation m/n=1/x(λ)+1/y(λ)+1/z(λ) with n=b+aλ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form x(λ)=kn(λ), y(λ)=n(λ)(s+rλ), z(λ)=(kl/r)(s+rλ) where parameters k,l,s,r∈ Z obey certain conditions depending on $a,b$. The conditions imply restrictions for some choices of $a,b$ which differ from the ones known in the case $m=4$. E.g., the modulus must be of the form $l(mk-1)$. One can also deduce that primes of the form $1+4K$ are excluded as modulus. Also if a=p≠ m is prime and $b=a+1$, i.e., n≡ 1 mod\, p, polynomial solutions are shown to be impossible. All results are valid for integers m ≥ 4.