It would help to know what year / model car this is.

I would start tracing back the P0501 error. VSS stands for Vehicle Speed
Sensor and is a little electronic pickup that goes where the old
speedometer cable used to be (on the differential). If the VSS does not
work, the speedometer would act up (you said yours does), and the tranny
would have a really hard time knowing what gear to select.

Without knowing year/model, this is fairly general, but my VSS ('94
Civic) has three terminals: Ground (black wire on my car), battery
voltage when ignition is on (Yellow w/ blue stripe) and pulse signal
back to the ECU (yellow w/ white stripe). If you have a multimeter you
can do some simple tests.

1. Unplug the connector at the VSS. Turn on ignition (do not start the
car). Measure if you have continuity between chassis ground and the
black wire at the connector. There should be continuity. Then measure
voltage between chassis ground and the yellow/blue wire. There should be
battery voltage. If either of these measurements don't give the expected
result there is a wiring problem somewhere.

2. Plug the connector back into the VSS. Set the hand brake, block the
front and back of the front passenger side tire. Put transmission in
neutral. Raise the front driver's side of the car so that the front left
tire is off the ground. Turn on ignition and measure voltage between
chassis ground and the yellow/white wire while you slowly rotate the
front left wheel. The voltage should alternate between something close
to 0V and something close to battery voltage (12 - 13 V typically).
Probing the yellow/white wire with the connector plugged in can be
tricky. I sometimes use a sewing needle to poke down into the connector
to reach the metal. If this test fails, you have a bad VSS.

All the above references are to a '94 Civic. Your wire colors may be
different. If you have AWD / 4WD and the VSS is on the rear differential
(don't know if it is on the rear differential or not, but IF it is),
elevate the rear wheel instead of the front.

If you replace the VSS, be aware that there is a tiny drive link in
there that may get lost if you are not careful.

I would sort out the VSS problem first, as you know this is a "real"
problem. Then I would reset the ECU and drive for a few days and hope
that the other error codes go away too.

Oxygen Sensor Heater Circuit

System TOO Lean (Bank 1)

Oxygen Sensor Heater Circuit

"A high voltage problem in the Electrical Load Detector"  -- 96
Acura Integra manual but expect most the Hondas of that vintage are
the same.

I'll take a swing at it.

Sounds like 2 interrelated problems.  One Emissions/Fuel and the other
VSS.

I'd say start by testing the VSS.  The primary control signals for a
transmission are vehicle speed and load. (VSS and MAP/Load/ELD?).. If
the input from either is missing or bad, you will have a very confused
transmission.  It may not be getting, or be getting bad information
from either or both..

It could also be that the VSS or load signal is causing it not to
supply enough fuel and that might also do it.

It could also be a computer or wiring that has failed.

You may want to take this to a professional.

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