Knowing all the values of resistances, capacitances and the value of the current source is there a way to find out how current is split in the two branches?
With a DC current source, the capacitors have no effect on the current division. The current will divide in accordance with well known circuit laws. These require that the current in each leg be inversely proportional to the resistance of that leg. This is because the voltage across each leg is the same since they are in parallel. Inspection of your circuit shows a resistance of over 5000 ohms in the left leg versus only 17 ohms in the right leg. Thus, be inspection without any calculations, one can state that most of the current will flow in the right leg.
If the circuit is AC, then you have to analyze it using Laplace transform or using Phasor. You will find the details in the book "Fundamentals of Electric Circuit by Alexander & Sadiku". There're two chapters related to the application of Laplace and Fourier transform.
And if the circuit is DC, after a short amount of time there would be no current through the capacitor. And if you need to consider the initial condition (just at the moment when the power is supply connected), then there's another chapter related to this in the book named "1st and 2nd order circuit".
The book of Alexander is the best engineering book I've ever read.