Could someone please explain if such integers exist and how to find them? If not, could someone please explain how to prove that they don't exist? Thank you!
They don't. The LHS is divisible by $3$ whereas the RHS isn't.
$\ Hint: $ If such integer exist, then $$6r + 15s + 9t = 1$$
so $$3\,(2r + 5s + 3t) = 1,$$ and this is an absurd.