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The $\textit{Gaia}$ Ultra-Cool Dwarf Sample $-$ III: Seven new multiple systems containing at least one $\textit{Gaia}$ DR2 ultra-cool dwarf

We present ten new ultra-cool dwarfs in seven wide binary systems discovered using $\textit{Gaia}$ DR2 data, identified as part of our $\textit{Gaia}$ Ultra-Cool Dwarf Sample project. The seven systems presented here include an L1 companion to the G5 IV star HD 164507, an L1: companion to the V478 Lyr AB system, an L2 companion to the metal-poor K5 V star CD-28 8692, an M9 V companion to the young variable K0 V star LT UMa, and three low-mass binaries consisting of late Ms and early Ls. The HD 164507, CD-28 8692, V478 Lyr, and LT UMa systems are particularly important benchmarks, because the primaries are well characterised and offer excellent constraints on the atmospheric parameters and ages of the companions. We find that the M8 V star 2MASS J23253550+4608163 is $\sim$2.5 mag overluminous compared to M dwarfs of similar spectral type, but at the same time it does not exhibit obvious peculiarities in its near-infrared spectrum. Its overluminosity cannot be explained by unresolved binarity alone. Finally, we present an L1+L2 system with a projected physical separation of 959 au, making this the widest L+L binary currently known.