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We introduce a two-parameter ensemble of generalized $2\times 2$ realsymmetric random matrices called the $\beta$-Rosenzweig-Porter ensemble(\brpe), parameterized by $\beta$, a fictitious inverse temperature of theanalogous Coulomb gas model, and $\gamma$, controlling the relative strength ofdisorder. \brpe\ encompasses RPE from all of the Dyson's threefold symmetryclasses: orthogonal, unitary and symplectic for $\beta=1,2,4$. Firstly, westudy the energy correlations by calculating the density and 2nd moment of theNearest Neighbor Spacing (NNS) and robustly quantify the crossover amongvarious degrees of level repulsions. Secondly, the dynamical properties aredetermined from an exact calculation of the temporal evolution of the fidelityenabling an identification of the characteristic Thouless and the equilibrationtimescales. The relative depth of the correlation hole in the average fidelityserves as a dynamical signature of the crossover from chaos to integrabilityand enables us to construct the phase diagram of \brpe\ in the $\gamma$-$\beta$plane. Our results are in qualitative agreement with numerically computedfidelity for $N\gg2$ matrix ensembles. Furthermore, we observe that for large$N$ the 2nd moment of NNS and the relative depth of the correlation holeexhibit a second order phase transition at $\gamma=2$.

Dynamical Signatures of Chaos to Integrability Crossover in $2\times 2$ Generalized Random Matrix Ensembles