  We investigate how symmetries and conserved quantities relate to the
occurrence of the boundary time crystal (BTC) phase in a generalized spin model
with Lindblad dissipation. BTCs are a non-equilibrium phase of matter in which
the system, coupled to an external environment, breaks the continuous time
translational invariance. We perform a detailed mean-field study aided by a
finite-size analysis of the quantum model of a p,q-spin-interaction system, a
generalized p-spin interaction system, which can be implemented in
fully-connected spin-1/2 ensembles. We find the following conditions for the
observation of the BTC phase: First, the BTC appears when the discrete symmetry
held by the Hamiltonian, $\mathbb{Z}_2$ in the considered models, is explicitly
broken by the Lindblad jump operators. Second, the system must be coupled
uniformly to the same bath in order to preserve the total angular momentum
during the time evolution. If these conditions are not satisfied, any
oscillatory behavior appears only as a transient in the dynamics and a
time-independent stationary state is eventually reached. Our results suggest
that these two elements may be general requirements for the observation of a
stable BTC phase relating symmetries and conserved quantities in arbitrary spin
models.
