Explore the intricate connections between five-dimensional BPS spectra and discrete integrable systems in this advanced physics lecture. Delve into the mathematical foundations and theoretical implications of exactly solvable models as presented by Fabrizio Del Monte. Gain insights into cutting-edge research at the intersection of quantum field theory, string theory, and mathematical physics. Examine the role of discrete integrable systems in understanding the structure of BPS states in five-dimensional theories. Analyze the methods and techniques used to derive exact results in this complex field of study.