  When a continuous symmetry is spontaneously broken in nonrelativistic
systems, there appear either type-I or type-II Nambu-Goldstone modes (NGMs)
with linear or quadratic dispersion relation, respectively. When equation of
motion or the potential term has an enhanced symmetry larger than that of
Lagrangian or Hamiltonian, there can appear quasi-NGMs if it is spontaneously
broken. We construct a theory to count the numbers of type-I and type-II
quasi-NGMs and NGMs, when the potential term has a symmetry of a non-compact
group. We show that the counting rule based on the Watanabe-Brauner matrix is
valid only in the absence of quasi-NGMs because of non-hermitian generators,
while that based on the Gram matrix [DT & MN, arXiv:1404.7696, Ann. Phys. 354,
101 (2015)] is still valid in the presence of quasi-NGMs. We show that there
exist two types of type-II gapless modes, a genuine NGM generated by two
conventional zero modes (ZMs) originated from the Lagrangian symmetry, and
quasi-NGM generated by a coupling of one conventional ZMand one quasi-ZM, which
is originated from the enhanced symmetry, or two quasi-ZMs. We find that,
depending on the moduli, some NGMs can change to quasi-NGMs and vice versa with
preserving the total number of gapless modes. The dispersion relations are
systematically calculated by a perturbation theory. The general result is
illustrated by the complex linear $O(N)$ model, containing the two types of
type-II gapless modes and exhibiting the change between NGMs and quasi-NGMs.
