  We explore a novel class of Yang-Baxter deformed AdS$_{4}$ $\times$ CP$^{3}$
backgrounds [Jour. High Ener. Phys. \textbf{01} (2021) 056] which exhibit a
non-chaotic dynamics for (super)strings propagating over it. We explicitly use
the \textit{Kovacic's algorithm} in order to establish non-chaotic dynamics of
string $ \sigma $ models over these deformed backgrounds. This analysis is
complemented with numerical techniques whereby we probe the classical phase
space of these (semi)classical strings and calculate various chaos indicators,
such as, the Poincar\'{e} sections and the Lyapunov exponents. We find
compatibility between the two approaches. Nevertheless, our analysis does not
ensure integrability; rather, it excludes the possibility of non-integrability
for the given string embeddings.
