  A nut graph is a simple graph whose adjacency matrix has eigenvalue~0 with
multiplicity~1 such that its eigenvector has no zero entries. Motivated by a
question of Fowler et al.~[\emph{Disc. Math. Graph Theory} 40 (2020), 533--557]
to determine the pairs $(n,d)$ for which a vertex-transitive nut graph of order
$n$ and degree $d$ exists, Ba\v si\'c et al.\ [\arxiv{2102.04418}, 2021]
initiated the study of circulant nut graphs. Here we first show that the
generator set of a circulant nut graph necessarily contains equally many even
and odd integers. Then we characterize circulant nut graphs with the generator
set $\{x,x+1,\dots,x+2t-1\}$ for $x,t\in\mathbb{N}$, which generalizes the
result of Ba\v si\'c et al.\ for the generator set $\{1,\dots,2t\}$. We further
study circulant nut graphs with the generator set
$\{1,\dots,2t+1\}\setminus\{t\}$, which appears to yield nut graphs of every
even order $n\geq 4t+4$ whenever $t$~is odd such that $t\not\equiv1\pmod{10}$
and $t\not\equiv15\pmod{18}$. This observation is tested here computationally
for all $t\leq1300$, and it fully resolves Conjecture~9 from Ba\v si\'c et
al.~[\arxiv{2102.04418}, 2021]. We also study existence of $4t$-regular
circulant nut graphs for small values of~$t$, which partially resolves
Conjecture~10 of Ba\v si\'c et al.~[ibid.].
