Epilepsy is a long-term neurological condition marked by uncontrollable seizure activity. Electroencephalography (EEG) is widely used to diagnose this disease. The creation of algorithms for automated processing and analysis of EEG data represents a crucial research domain within the fields of machine learning and artificial intelligence. In this study, we present a hybrid binary classification model that combines a Long Short-Term Memory (LSTM) network with a transformer for the automatic detection of epilepsy-related abnormalities. A hybrid approach, in which several heterogeneous models are combined into one, is a promising approach in the realm of contemporary machine learning, since it allows one model to take advantage of different algorithms. Nonlinear characteristics of the electroencephalogram signal, such as the Hurst exponent, Higuchi fractal dimension, detrended fluctuation analysis (DFA), sample entropy, the highest Lyapunov exponent, correlation dimension, autocorrelation, statistical characteristics of the signal, along with frequency characteristics obtained using the fast Fourier transform in five frequency ranges, most often used in EEG analysis, were employed as informative features for the suggested classification model. In the conducted computational experiments, the proposed model achieved high performance indicators on the test sample: 94.6% sensitivity, 99.8% specificity, 0.971 F1-score and 0.972 AUC.