The recent advances in high-throughput single-cell sequencing have created an urgent demand for computational models which can address the high complexity of single-cell multiomics data. Meticulous single-cell multiomics integration models are required to avoid biases towards a specific modality and overcome sparsity. Batch effects obfuscating biological signals must also be taken into account. Here, we introduce a new single-cell multiomics integration model, Single-cell Multiomics Autoencoder Integration (scMaui) based on variational product-of-experts autoencoders and adversarial learning. scMaui calculates a joint representation of multiple marginal distributions based on a product-of-experts approach which is especially effective for missing values in the modalities. Furthermore, it overcomes limitations seen in previous VAE-based integration methods with regard to batch effect correction and restricted applicable assays. It handles multiple batch effects independently accepting both discrete and continuous values, as well as provides varied reconstruction loss functions to cover all possible assays and preprocessing pipelines. We demonstrate that scMaui achieves superior performance in many tasks compared to other methods. Further downstream analyses also demonstrate its potential in identifying relations between assays and discovering hidden subpopulations.