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We study an integrable spin chain with three spin interactions and thestaggered field ($\lambda$) while the latter is quenched either slowly (in alinear fashion in time ($t$) as $t/\tau$ where $t$ goes from a large negativevalue to a large positive value and $\tau$ is the inverse rate of quenching) orsuddenly. In the process, the system crosses quantum critical points andgapless phases. We address the question whether there exist non-analyticities(known as dynamical phase transitions (DPTs)) in the subsequent real timeevolution of the state (reached following the quench) governed by the finaltime-independent Hamiltonian. In the case of sufficiently slow quenching (when$\tau$ exceeds a critical value $\tau_1$), we show that DPTs, of the formsimilar to those occurring for quenching across an isolated critical point, canoccur even when the system is slowly driven across more than one critical pointand gapless phases. More interestingly, in the anisotropic situation we showthat DPTs can completely disappear for some values of the anisotropy term($\gamma$) and $\tau$, thereby establishing the existence of boundaries in the$(\gamma-\tau)$ plane between the DPT and no-DPT regions in both isotropic andanisotropic cases. Our study therefore leads to a unique situation when DPTsmay not occur even when an integrable model is slowly ramped across a QCP. Onthe other hand, considering sudden quenches from an initial value $\lambda_i$to a final value $\lambda_f$, we show that the condition for the presence ofDPTs is governed by relations involving $\lambda_i$, $\lambda_f$ and $\gamma$and the spin chain must be swept across $\lambda=0$ for DPTs to occur.

Tuning the presence of dynamical phase transitions in a generalized $XY$ spin chain