Order Payment Method and Delivery REST API

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I am currently working on an app for a client in shopify and he would like to have specific payment method on the app. We are using the REST API to get all the information but when we create order's we can't set the delivery method or the payment method. We don't want to use shopify Checkout API but instead offer our own solution in the app payment system. Is these possible and if so how?

Thanks in advance.