  Given a $G$-invariant potential $\mathcal{V}$ of a scalar multiplet
$\varphi$, there may exist a set of homogenous linear equations that constrain
the components of a stationary point of $\mathcal{V}$ independently of the
coefficients of the terms in $\mathcal{V}$. We call them homogeneous linear
intrinsic constraints (HLICs). HLICs in a stationary point manifest as HLICs in
the corresponding vacuum alignment of $\varphi$, which plays a central role in
predictive phenomenological models. We discover that a group $\tilde{H}$
generates HLICs if the terms in $\mathcal{V}$ satisfy a condition, which we
call the compatibility condition. In this paper, we also develop a procedure,
which involves splitting $\mathcal{V}$ into smaller parts, to establish the
existence of specific stationary points using arguments based on symmetries
without the need for explicitly extremizing the potential. Using this
procedure, we obtain $\tilde{H}$ as a direct product of the symmetry groups
associated with the various irreducible multiplets (irreps) in $\varphi$. This
results from considering the potentials of the irreps separately and verifying
if the cross terms are compatible with $\tilde{H}$.
