In order to connect an eCommerce website which is built in Magento 1.X to a payment provider, I know I can:
Select from existing extension (plugin) that will connect me to well known payment providers, as available in this marketplace.
Add payment method that from existing list that defined in Magento framework, as described here
but I am looking for a 3rd option: to integrate a website that is implemented in Magento 1.X a to a payment provider (PSP) that is not part of the predefined ones and has no ready-made extension that I only need to customize.
Questions:
There is no restriction, that does not allow you to implement your own payment service from Magento. You can develop your own module to connect the PSP's API. Therefor, the PSP must have an API. Note, that payments are one of the most sensitive features in eCommerce which requires indepth-knowledge on both Magento and SPS-side. Hardening required :-) Maybe it would be easier to switch the payment gateway to a provider which already has a Magento solution?