I really have to say sorry for the late reply. I have been busy with other topics, so I was not really able to dig deeper into the repair.
I recognized powering on the device, while the filter box is unscrewed and connected via the cables is not a good idea. There is connection screw, which is not only designed for mechanical purposes. It seems to serve in addition as star ground connection. The readings went all over the place, but except that the LIA is not damaged.
The lock-in amplifier has at least two oscillators, I can locate. One oscillator should be an analog VCO (for the internal reference oscillator) and the other one a crystal oscillator for the MCU. The frequency of the internal reference oscillator is digitally adjusted with buttons on the front panel. Based on this adjustments, the MCU seems to change the output value of a DAC. The "actual" frequency (read back via the MCU) can be displayed on the LCD. The difference between both frequency was rather high, so you are absolutely right, they are not locked.
The analog VCO seems to work properly - the OSC output has shown a plausible signal up to the upper frequency limit and the unlock condition occures with the internal and an external source.
I have seen an ADI part close to the area, where I assume the PLL, VCO and the phase shifter for the reference input is located - I guess, this could be the DAC for the phase-shift. Maybe this could be a good starting point for further tests.
@LCR: Hi, I have read this document, but there is no schematic included - just a block diagram. I have seen a service manual for the 5210 (should be pretty much the same) on ebay a couple of weeks ago. The seller has shown a pictures which included a schematic.