  We consider the Ising Curie--Weiss model on the complete graph constrained
under a given $\ell_{p}$ norm for some $p>0$. For $p=\infty$, it reduces to the
classical Ising Curie--Weiss model. We prove that for all $p>2$, there exists
$\beta_{c}(p):=\frac{3}{p^{2/p}}\cdot\tfrac{\Gamma(1+1/p)}{\Gamma(1+3/p)}$ such
that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and
satisfies an appropriate Gaussian CLT; whereas for $\beta>\beta_{c}(p)$ the
magnetization is not concentrated at zero similar to the classical case. Though
$\beta_{c}(2)=1$, we have $\beta_{c}(p)>1$ for $p>2$ and
$\lim_{p\to\infty}\beta_{c}(p)=3$. We further generalize the model for general
symmetric spin distributions and prove a similar phase transition. For $0<p<1$,
the log-partition function scales at the order of $n^{2/p-1}$. The proofs are
mainly based on the generalized Hubbard--Stratonovich (GHS) transform, which is
of independent interest.
