Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$
is
$$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\delta_{1}\left(m,n\right)}\right)$$
and that the number of *primitive* $m$-dimensional subgroups of $\mathbb{Z}^{n}$
of covolume $\leq X$
is
$$c_{2}\left(m,n\right)X^{n}+O\left(X^{n-\delta_{2}\left(m,n\right)}\right)$$
for some positive $c_{1}\left(m,n\right)$, $c_{2}\left(m,n\right)$,
$\delta_{1}\left(m,n\right)$ and $\delta_{2}\left(m,n\right)$
whose specific values are not relevant to my question.

I refer to an orthogonal transformation with determinant 1 that is neither plus or minus the identity, as a *non-trivial* rotation.

My question is: what can be said about the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ (primitive or not) of covolume $\leq X$ that satisfy that they are invariant under a non-trivial rotation of their spanned subspace? I am looking for an upper bound, and am only interested in the exponent (and not in the constant).

Two remarks:

When considering the similarity classes of $m$-dimensional lattices up to rotation and rescaling, these classes are identified with the points of the space $$SL_m(\mathbb{Z})/SL_m(\mathbb{R})\setminus SO_m(\mathbb{R}).$$

The classes of the lattices which are invariant under non-trivial rotations fall in the boundary of this space (when identified with a suitable fundamental domain inside $SL_m(\mathbb{R})\setminus SO_m(\mathbb{R})$). This should hint at the fact that the number of these lattices is $o\left(X^{n}\right)$; indeed, we were able to show that it is $O\left(X^{n-\frac{1}{n-1}}\right)$, but we expect it is actually $O\left(X^{n-1}\right)$, or even below that. For example, for $n=3$ and $m=2$, this number is finite regardless of $X$.

I am more than happy to know the answer for just $m=n-1$.